EP2777159A1 - Method and system for coding information - Google Patents

Abstract

A method and an apparatus are described for coding information, the method comprising obtaining a list of integers to be coded; determining a hyperspatial pyramid having a size adapted for coding the list of integers, the hyperspatial pyramid having a plurality of vertices, the number of which is determined by the degree of the hyperspatial pyramid which is equal to the sum of the integers of the list of integers and by the dimension of the hyperspatial pyramid which is equal to the number of integers of the list of integers minus one; indexing the list of integers in this pyramid by means of an indexation system; and providing an indication of the indexation in the hyperspatial pyramid.

Description

 METHOD AND SYSTEM FOR CODING INFORMATION Cross Reference to Corresponding Requests

This patent application claims the priority date created by the filing of Canadian Application No. 2,758,243, the title of which is "Method and apparatus for encoding information" which was filed on November 10, 201 1.

Application domain

This invention relates to the field of coding. More specifically, this innovation relates to a method and a system for encoding information using a hyperspace pyramid. Prior art

Let x ₀ , ..., _{n be} integers from a list of integers and X be the generated integer; to encode a list of natural numbers into a single integer and its inverse return to finding coding and decoding functions ensuring the following correspondence:

Code (x ₀ , ..., _n ) → X Decode (X) → x ₀ , ..., _n An encoding system consists in establishing a correspondence law called code between the information to be represented and the numbers obtained, of so that each information corresponds to one and usually a single number from which it will be possible to retrieve the initial information. Some coding strategies rely on functional bijective mechanisms contained, for example, in the fundamental theorem of arithmetic or in polynomial generation methods of numbers. In the first case, the uniqueness of the decomposition of a number into a product of prime factors is the key to the bijective mechanism. In the second case, the uniqueness of the decomposition of a number into a sum of elementary numbers must be established beforehand.

For example, the Fibonacci encoding makes it possible to concatenate a list of integers into a binary number such that the separators between the integers are represented by the binary pattern 1 1. Fibonacci ([\, 2,3,9,8,7]) → {1 101 1001110001 10000110101} ₂ → {57088107}

n ^ o ^ io ^ T ^ o ^ oooi

The person versed in the field will appreciate that in general, beyond the theoretical choice of the coding method, the length of the list to be coded and the size of the whole numbers to be treated must be considered as an essential problem from the point of view practical and application. The person versed in the domain will also appreciate that apart from the problem of compression of the list of integers, the size of the generated code, ie its number of representative digits, is a fundamental element in the choice of the methods of coding. Given the size of the data to be processed and the levels of algorithmic complexity encountered, performance collapse phenomena, particularly during the decoding phases, are practically immediate and disqualify certain theoretically interesting methods with regard to the initial coding phase. .

Since the various encoding families address a certain number of issues related for example to the topics of compression, transmission, structuring, or data encryption, the properties of the generated codes can in this case be very different. In indexing encoding systems, the law of correspondence is a relation of order which calculates the rank of a list or sequence of integers, and which makes it possible to classify or to sort this list starting from a unique integer which can in turn be decoded to restore the list.

The person versed in the field will appreciate that there are many possible applications to coding. In particular, the scheduling of certain sequences of integers is generally the primary goal of indexing coding methods. The sequences to be ordered are essentially combinatorial and are, for example, combinations or permutations that must be ordered. Extensive applications range from linear algebra, algebraic geometry, graph theory to lexicographic and alphabetic dictionary ranking, and the enumeration and positioning of combinations in game theory. Trellis indexing is particularly important in routing and parallel algorithm problems. Thus, the routing can be defined as a software or hardware mechanism that routes the information between the source and the destination during a communication in an incompletely connected network of static or dynamic type: it makes the choice of the paths when there are some many, he manages the conflicts, finally he ensures that the messages arrive well and in the right order.

The addressing of data and their storage areas in databases or data structures is at the basis of access to digital information and its structuring. The interest of coding by indexing the information is to free oneself from any physical organization of the memory, prior to the manipulation of the data. This physical organization is typically articulated around notions of tables or lists whose structure, size and memory location must be explicitly declared. In contrast, indexing coding privileges an implicit description of the information through the properties of the encoded number. For example, the representation of information in spreadsheets uses standard description formats that specify the organization conventions of the files. These formats, such as the Comma Separated Value, explicitly describe how information should be recorded.

The person versed in the field will further appreciate that the lattice indexing functions can be used as powerful hash functions. In order to optimize access to information, hash techniques allow from hash math functions to directly calculate the index of a table from an alphanumeric string called the key. The directly indexed memory zone then contains the information associated with the key or a new set of addresses allowing indirect access to these addresses.

The person versed in the field will also appreciate that electronic identification is increasingly used to ensure the traceability of objects, food or living beings. The different typologies of identifiers are the identifiers for the registration, for the indexing of the information: the primary key, or else for the connection. Index coding systems are particularly well suited to W these types of problem because they facilitate inter alia alphanumeric sorts and allow quick identification in databases.

Finally, the functions of encoding or mathematical coupling are often the basis of encryption systems and can be the basis of the one-to-one relationship between the encoding and decoding operation. The use of the properties of the coded numbers, for example the decomposition into a product of prime numbers, can also serve as an independent or complementary approach.

In positional encoding, positional numeration systems are used to encode and decode integer vectors from a counting base higher than the k - 1 cardinal of the encoding alphabet. The encoding of a vector (α ₀ , · · · α _κ ) in base k will then be

I _B = cc _Q .B ^k + a _v B ^k - ^l + - - - + a _k .B °

If for example the integers (1, 2, 3, 4, 5, 6, 7, 8, 9) are associated with the letters of the alphabet (A, B, C, D, E, F, G, H, I ), the following code will be obtained 2.10 ⁶ +1.10 ⁵ + 2.10 ⁴ + 2.10 ³ + 1.10 ² + 7.10 ¹ + 5.10 ° → {2122175} ₁₀ corresponding to the word

BABBAGE coded in base 10. In this case, the alphabet does not exceed the number of available digits, the code obtained corresponds to a simple substitution of the letters of the word by the associated digits. By enriching the alphabet, the polynomial calculation becomes necessary, the letters being able to be coded on more than one digit, for example in hexadecimal number with A = 10, 5 = 1 1 · · ·.

The person versed in the field will appreciate that the coding of a list of integers can be done from the modular integer representation system or the Residue Number System (RNS). The RNS system is defined from a set of prime numbers two by two named modules and noted {nto, m ;, · · ·, m "}. Therefore, GCD (mi, mj) = 1, with i ≠ j. An integer X can be then represented by a set of integers called residues {r _0, r _lt · · ·, r _n} such that x = X mod rri _j with <Xi <mi. This representation is, on the other hand, unique for X <M-1 with M ·

The coding principle is therefore to consider the list of integers to be coded as a list of residues and find the associated decimal representation. It is possible to find modules higher than the respective integers of the list, to use a sequence of successive prime numbers all higher than their respective residues or beginning with the prime number immediately greater than the maximum integer of the list to be encoded. The following example makes it possible to code the projective point [69, 78, 73, 71, 77, 65] in dimension 5 from the module 73. The Chinese coding function is essentially a function making it possible to solve a congruence system. from a list of integer residues.

Chinese ([69, 78, 73, 71, 77, 65], [73, 79, 83, 89, 97, 101]) → {252617134512} 252617134512 mo = 73 = 69 ^»» 252617134512 me 101 = 65

The decoding of the decimal number is not unique, it follows that it is necessary to know the triplet (Code, initial module, length of the list) to perform the decoding. The necessary triplet will be noted {252617134512} ( ₇ 3 ₅ ). The list could be encoded with an infinity of other triplets such as for example {412251249819} (79.5), 79 being in this case the first integer greater than 78.

Cantor's coding is associated with the notion of coupling function developed to answer the enumerability problem of natural numbers. The coupling function of Cantor is reduced to a polynomial of the second degree having the abscissa and the ordinate of the Cartesian plane as a variable. Figure 1 shows the order in which the polynomial associates the pairs of coordinates with the indices according to the following formula. For example I _K (1, 2) ^ 8. It is possible to obtain a second coupling polynomial by substituting x for y and in this case I _K (1, 2) -> · 7.

_ (x + y) * (x + y + l) _{t ^}

2 ^{+ y}

The variant illustrated in FIG. 1b presents a boustrophedonic variant.

The Cantor coupling function for encoding and decoding a pair of natural numbers has been extended to any number of integers in two ways different. The first is to group the integers by successively constituting pairs of pairs of integers. This method has the advantage of the simplicity of Cantor's initial coding function: a polynomial of degree 2 but generates polynomials of degree n ² . This exponential growth affects in particular the size of the calculated indices. An alternative to this approach is the n-dimensional extension of the coupling function. An expression based on binomial coefficients is presented by Cegielski and Richard in the following form:

(x, + ... + X. + k - \ x, + ... + x _k , + k - 2

K (x, ..., x _k ) = + + ·

k - l Unfortunately the hyperspatial coding of Cantor is not bijective. Vectors of different dimensions can generate the same code. This problem is also present with the successive coupling in pairs. In this case, one can conventionally add the arity of the vector to be encoded as additional coordinate.

In Ernesto Pascal's coding, an integer N can be represented in a unique way for some degree n as follows:

N =, x "> ^■ · ^■ > x ₂ > X _j > 0.

The number 1000 will for example be represented by the sequence {12,9,8,7,6,5,3} Cm corresponding to the first seven columns of Pascal's triangle or else by the sequence {19,8,3} _C ³ _ns corresponding to the first three columns. The coding therefore applies to strictly decreasing integer sequences and is used to solve the problem of lexicographic ordering of the r-combinations of Cl ^ 2 of {1,2, · · ·, r). The formula can be slightly modified to calculate the index of a given combination in the form of a sequence of increasing integers and it is obtained:

There is therefore a need for an information coding method which solves at least one problem of the prior art.

List of figures Figure la illustrates the Cantorian order while figure lb illustrates the Cantorian order boustrophedonique.

Figure 2 illustrates the De Gua triangle of Malves and its projective extension.

Figure 3 illustrates a pyramidal lattice and the labeling of projective coordinates.

Figure 4 illustrates an embodiment of the coding and decoding system. Figure 5 illustrates an implementation of the GLOBAL algorithm and an implementation of the LOCAL algorithm.

Figure 6 illustrates recursive trees for

Figure 7 illustrates the positional indexing for T _{3 2} in quaternary base.

Figure 8 illustrates the expansion of large integers. Figure 9 illustrates the numerical order for the T4.3 lattice.

Figure 10 illustrates an implementation of the NUMORD algorithm.

Figure 11 illustrates a pyramidal decomposition of a number. Figure 12 illustrates an implementation of the ORDNUM algorithm.

Figure 13 illustrates the lexicographic coding of the T4.3 lattice. Figure 14 illustrates an implementation of the LEXORD algorithm. Figure 15 illustrates an implementation of the ORDLEX algorithm.

Figure 16 illustrates the Projective Gray coding (4.3).

Figure 17 illustrates an implementation of the GRAORD algorithm.

Figure 18 illustrates the course of T4,3 in projective Gray. Figure 19 illustrates an implementation of the ORDGRA algorithm.

Figure 20 illustrates an implementation of the VN2VM algorithm and the VM2VN algorithm.

Figure 21 illustrates the combinatorial tree

(6

 Figure 22 illustrates the tree and the digital tags.

 v4 Figure 23 illustrates the tree and the lexicographic labels.

Figure 24 illustrates an example of a combinatorial projective subspace. Figure 25 illustrates an example of a bijection between projective points and combinations. Figure 26 illustrates an example of projective and binary generation of combinations. Figure 27 illustrates the Euclidean and affine coordinates of a Cantorian point. Figure 28 illustrates the taxonomy of orders. Figure 29 illustrates Cantorian paths.

Figure 30 illustrates a possible implementation of the method for encoding information.

Figure 31 illustrates spiral paths. Figure 32 illustrates an implementation of the SPIORDc algorithm and the ORDSPIc algorithm.

Figure 33 illustrates an implementation of the SPIORDa algorithm and the ORDSPIa algorithm. Figure 34 illustrates a spiral chain for T ₆₃ . Figure 35 illustrates a convolution mechanism.

Figure 36 illustrates convolutions of Vandermonde for

Figure 37 illustrates an implementation of the CRKREC algorithm.

Figure 38 illustrates an implementation of the XENUME algorithm and the YENUME algorithm.

Figure 39 illustrates an implementation of the COMBO algorithm.

Figure 40 illustrates the optimized calculation of the binomial coefficient.

Figure 41 illustrates an implementation of the PRODYN algorithm.

Figure 42 illustrates an implementation of the MATPRT algorithm. Figures 43a and 43b illustrate a matrix multiplication between columns.

Figure 44 illustrates a curve that represents for a given dimension the values of the binomial coefficients.

Figure 45 illustrates an implementation of the ENCDEC algorithm and the DIGBIN algorithm. Figure 46 illustrates an implementation of the NUMFCT algorithm. Figure 47 illustrates an implementation of the DRVREC algorithm. Figure 48 illustrates an implementation of the DRVSYM algorithm.

Figure 49 illustrates an implementation of the BINRP4 algorithm.

Figure 50 illustrates an implementation of the BINRPNi algorithm and RPNBINi algorithm. Figure 51 illustrates an implementation of the GLOBALo algorithm and the LOCALo algorithm.

Figure 52 illustrates an implementation of the ORDNUMo algorithm and the ORINUMo algorithm.

Figure 53 illustrates an implementation of the INDCOM algorithm and the COMIND algorithm.

Figure 54 illustrates an implementation of the MPLNUM algorithm. Summary

According to one aspect of the invention, a method is described for encoding information, the method comprising obtaining a list of integers to be encoded; determining a hyperspatial pyramid having a size suitable for encoding the list of integers, the hyperspace pyramid having a plurality of vertices the number of which is determined by the degree of the hyperspace pyramid which is equal to the sum of the integers of the integer list and by the dimension of the hyperspatial pyramid which is equal to the number of integers in the list of integers minus one; index the list of integers in this pyramid by means of an indexing system; and providing an indication of indexing the list of integers in the hyperspace pyramid to thereby encode said list of integers to be encoded.

According to one embodiment of the method described above, the method further comprises providing a list of alphanumeric codes and transforming the list of alphanumeric codes into a list of integers to code. According to an embodiment of the method described above, the indexing system makes it possible to index each of the plurality of vertices of said hyperspace pyramid. According to an embodiment of the method described above, the indication of the indexation of the list of integers in the hyperspace pyramid comprises an integer representing the index of the projective point in the indexing system, m the total degree of said list of integers and n the number of integers in the list of integers. According to one embodiment of the method described above, the method further comprises transforming the integer representing the index of the projective point by mx and nx combinatorial coordinates.

According to an embodiment of the method described above, the method further comprises the transformation of m the total degree of the list of integers and n the number of integers of the list of integers by an integer Y representing a global index .

According to an embodiment of the method described above, the method further comprises summing the integer Y representing a global index to the integer representing the index of the projective point to provide a total index Z, the indication of indexing in the hyperspace pyramid comprising the total index Z. According to an embodiment of the method described above, the list of integers is obtained from a file.

According to one embodiment of the method described above, the step of providing indexing indication in the hyperspace pyramid includes storing said indexing indication in a file. According to an embodiment of the method described above, the step of providing the indication of the indexation of the list of integers in the hyperspace pyramid comprises the transmission of said indication of the indexation of the list of integers to a processing unit via a network.

According to an embodiment of the method described above, the method is used to code a list of relative integers, the method comprising providing the list of relative integers and generating the list of integers by means of the list of integers . According to an embodiment of the method described above, the step of providing the indication of the indexing of the list of integers in the hyperspace pyramid comprises displaying said indication of the indexing of the list of integers. on a screen.

In another embodiment, the method described above is used to compress a string in which the string is associated with said list of integers.

In another embodiment, the method described above is used to code a list of lists, the method comprising the iterative use of the method described above for encoding a list of lists. In another embodiment, the method described above is used to encrypt a list of integers, in which the indexing of the list of integers corresponds to the encryption of the list of integers.

According to a broad aspect of the invention a method is described for decoding coded information by means of the method for coding information described above, the method comprising obtaining an indication of the indexation of the list of whole in the hyperspace pyramid; obtaining an indication of an indexing system used to generate said indication of the indexing of the integer list in the hyperspace pyramid and determining by indexing and indicating an indexing system used said list of integers. According to a broad aspect of the invention, an information storage device is described for storing executable instructions by a processor, which when executed will cause the processor to execute a method for encoding information. the method comprising obtaining a list of integers to be encoded; determining a hyperspatial pyramid having a size suitable for encoding the list of integers, the hyperspace pyramid having a plurality of vertices the number of which is determined by the degree of the hyperspace pyramid which is equal to the sum of the integers of the integer list and by the dimension of the hyperspatial pyramid which is equal to the number of integers in the list of integers minus one; index the list of integers in this pyramid by means of a system indexing; and providing an indication of indexing the list of integers in the hyperspace pyramid to thereby encode said list of integers to be encoded.

According to a broad aspect of the invention, a computer is described, the computer comprising a display unit; a processor; a data storage unit comprising a program configured to be executed by the processor, the program for encoding information, the program comprising instructions for obtaining a list of integers to be encoded; instructions for determining a hyperspatial pyramid having a size suitable for encoding the integer list, the hyperspace pyramid having a plurality of vertices the number of which is determined by the degree of the hyperspace pyramid which is equal to the sum of the integers in the list of integers and by the dimension of the hyperspatial pyramid which is equal to the number of integers in the list of integers minus one; instructions for indexing the list of integers in this pyramid by means of an indexing system; instructions for providing an indication of indexing the list of integers in the hyperspace pyramid to thereby encode said list of integers to be encoded. An advantage of the system described is that it is a multi-in-one polymorphic system because it is a combined system of coding and indexing but also of transcoding between combinatorial coding systems using the coefficients binomial.

An advantage of the system described is that it is intended for encoding / indexing / transcoding lists of any integers without predefined combinatorial structures as well as for particular lists with predefined combinatorial structures such as polynomial coefficient lists or lists of combinations without repetitions.

Another advantage of the system described is that it allows to work in bijective encoding mode Code (X) → Y and Decode (Y) → X or in encryption mode Code (X) → Y Decode (Y, key 1, key 2) → X. Another advantage of the system described is that it can provide a solution to various problems such as the nature and size of the data, the size of the coding / decoding program, the size of the generated code and the properties of the generated code. Thus, with regard to the nature and size of the data, the method described may allow the use of structured or non-structured data of any size.

As far as the size of the coding / decoding program is concerned, it is possibly implementable by means of a few dozen instructions and a standard multiprecision integer processing library, such as for example GMP.

As far as the size of the generated code is concerned, there is no explosion of size, whether in compressed mode or not, because there is no factorization by prime numbers or calculations using the exponentials.

In terms of coding and decoding performance, it may be possible to use optimized algorithms using next-generation algorithms for calculating and finding binomial coefficients in Pascal's triangle.

With regard to the properties of the generated code, the generated code can be indexed multi-use for for example at least one of the scheduling of the integer lists, the routing architectures, the addressing and the hashing of data. electronic identification and cryptography.

detailed description

The person versed in the field will appreciate that Pascal's triangle has many properties. In particular, for the enumeration of the monomials of a homogeneous polynomial of degree m in dimension n:

It should also be noted that Cr (m, n) = Cr (n, m) and consequently the writing of the central binomial coefficient of the line r, Cr (r, r). Moreover, the sum of the binomial coefficients of this line has the value: The computation of a binomial coefficient from the binomial coefficients located in the adjacent column of left or then in the diagonale just superior is respectively: Moreover, the formula by recurrence calculating a binomial coefficient starting from its two superior neighbors makes it possible to build naively the triangle of Pascal and expresses: with r≥l.

Pyramidal numbers are figured numbers, ie numbers decomposed and visualized from points distributed on regular polytopes. Pascal's triangle is formed from triangular, tetrahedral and generally hyperte- torric numbers.

The disclosed system is based on the idea of using the combinatorial properties of the Pascal triangle and the algebraic properties of the hyperspace projective extensions of the De Gua triangle of Malves as a general indexing and coding system of integer lists. natural. For this, the one-to-one correspondence between the monomials of a polynomial P _mn and the points of a lattice T _mn will be used. The ordering of the monomials of this polynomial will consequently be equivalent to the ordering of the points of this lattice. Figure 2 shows the transition between the original triangle described by De Gua and its two-dimensional projective extension for a third-degree curve.

Definition and properties of pyramidal lattices

It will be appreciated by the person versed in the field that directly related to the notion of pyramidal number, a pyramidal lattice is as much a combinatorial structure as a hyperspace geometrical structure whose study concerns the algebraic geometry. The lattices are in the form of hyperspace pyramids, such as the tetrahedron of FIG. 3 or the triangle of FIG. 2. A tetrahedral pyramidal lattice, denoted T _mn corresponding to the line m + n and to the column m of the Pascal triangle, is associated with each binomial coefficient and therefore a hyperte- toric projective configuration of dimension n in homogeneous points, linearly

dependent. The points of the lattice are organized in a combinatorial way, in sub-lattices of dimension n - \ and of degree from 0 to m - \ (rule of the summation of the diagonal). Each lattice is located in Pascal's triangle by the pair (m, n) or by the summation of all the points of all the preceding lattices, which we will name the global index. The number of points, the associated pyramidal number, is equal by definition to the number of monomials of a homogeneous polynomial of degree m in n dimensions. The homogeneous projective coordinates of the points of a lattice T _mn can therefore be interpreted as the exponents of the monomials of an associated polynomial of degree m and of dimension n.

Each square of Pascal's triangle corresponds to a hyperspace pyramid whose number of vertices is equal to the number indicated in the box.

Order of monomials of homogeneous polynomials

A monomial ^{x "in} * ^o> * ^i" - ^"** _is defined as the product ^{Λ" Λ "°" Λ¾} and the total degree of _i is def ned as \ ™ \ = ™ "™ + _x + ,. .., + m _{k ή} vm = (m ^ m ^ m,) _{? represents} the vector of the exponents, and ^x ° ' ^l "' ^{k /} , represents the vector of the indices of the variables with ⁿ o ^{≤ n} i- ^≤n k _

The Euclidean definitions of the orders of the monomials of the nonhomogeneous polynomials are based on the notion of lexicographic order and of total degree of the monomials, ie the sum of the exponents of the variables of the monomials. By combining these two notions, we obtain in particular, the pure lexicographic order, the total and then the lexicographic order, the total order then the inverse lexicographic one. In the case of homogeneous polynomials, all monomials have by definition the same total degree and the ordering of the monomials can be realized from two ducal projective lexicographic orders, one on the exponents and the other on the indices, the monomials being represented in tensorial notation. In this case, inverse lexicographical orders are simple upward or downward variants of lexicographical orders.

Will be named numerically lexicographical order on exhibitors as a ^{vm vm} ^ mm fi _s j j _{and § seu} i _ent the far left nonzero value ^{Vm "νϊΗβ} is negative.

Be appointed numlex order numerical order on exhibitors such that vm = (m _k, m _k _ ..., m _{0) 0} ≥M _ayec m ... _{x k} ≥M

The combinatorial order will be the numerical order on the exponents such that vm = ( _mk , _mk ^, ..., m ₀ ) m ₀ > m _l ...> m _i

The lexicographic order of the indexes of variables such as ^{Vfla k ν β} if and if only the non-zero most left value of ^vn " ^νΗβ is negative will be named lexicographic order.

If the De Gua de Malves polynomial of figure 2 corresponding to a Euclidean curve of the third degree is ordered in the lexicographic order of its coefficients, it is obtained: a, by, ex, ey ² , fxy, gx ² , hy ^3, IXY ^2, kx ² .y, 2.x ^3. After substituting x for x0 / x2 and y for xl / x2, reduce the set of terms to the same denominator and finally keep the numerators, the homogeneous representation of the polynomial is obtained: IX _Q.

Noting vm _a = (3,0,0), vm _b = (2,1,0) ... vm, = (0,0,3), it is possible to obtain in reverse numerical order and in order the following sequences of monomials: (a, b, c, e, f, g, h, i, k, l)

(l, k, i, h, g, f, e, c, b, e).

Noting ν " _α = <2,2,2>, vn _b = (1,2,2) ... vn _/ = <0,0,0>, it is obtained in lexicographic order: (l, k, g, i, f, c, h, e, b, a). Principles of combinatorial coding

By using the combinatorial and projective properties of lattices, it is possible to define a general system for indexing the monomials of polynomials and simultaneously to use it as a coding system for lists of natural numbers: coordinates. It will be appreciated by the person skilled in the art that by transposing the integers into arbitrary alphanumeric codes, the encoding system can be extended to the encoding of text and alphanumeric character strings.

The system disclosed is therefore based on the following principles:

1. At each point of a lattice (m, n), also called the vertex of a hyperspace pyramid, is associated a vector of homogeneous coordinates of type [a ₀ : a _l : ...: a _n ]. The sum of the coordinates is equal to m. The position of a square in Pascal's triangle and consequently of the hyperspace pyramid and the associated hypersurface is given by the line and column of Pascal's triangle. The lines and columns associated with a square of Pascal's triangle correspond to the algebraic degrees and dimensions of the associated polynomials. Each vertex of the same pyramid: its support pyramid, has a signature formed of an equal number of coordinates whose sum represents the degree and the length the dimension minus one. This signature can be interpreted in combinatorial theory either as the projective coordinates of the vertex point belonging to the pyramid, or as the powers of the monomials of the polynomial representing the associated hypersurface, or as indices of the variables composing the monomials of this same polynomial. 2. The vector of homogeneous coordinates of type [a _t) : a _x : ...: a _n \ is considered as the vector of exponents of a monomial of total degree m, belonging to a polynomial of degree m in n dimensions .

3. Each trellis is associated with a travel order, also called indexing indication, points on particular Hamiltonian paths passing through all the points of the trellis according to one embodiment of the invention.

4. The local index of any monomial is considered as the index of the homogeneous point associated, that is to say the number of the point in the order of course of the chosen path.

5. According to one embodiment of the invention, any coordinate vector is coded or decoded in local mode thanks to the triplet (local index, total degree of the vector, dimension

("+!)) ^■

6. According to one embodiment of the invention, any coordinate vector is coded or decoded, in global mode, by virtue of the total index obtained by adding the local index and the global index. 7. According to one embodiment of the invention, the notion of coordinate vector is extended to any set of natural integers representing arbitrary alphanumeric codes.

8. The coding is to find bijective systems of numbering or indexing of the messages by double-numbering first in the Pascal triangle, the hyperspatial pyramid associated with the given message: the support pyramid, then the point -statement of the pyramid whose signature represents the message. The integer or tuple, usually the triplet of integers, resulting from this double numbering represents the encoded message in indexed form. The coding can be based on different numbering and vertex-point routing methods on the same support pyramid, and the resulting code differs from one method to another. 9. The decoding is to find from the code the support pyramid of the message and the signature of the vertex-point belonging to the pyramid. This double location can only be done by knowing a priori the method of course chosen for the numbering of the vertex-points on the support pyramid. Thus and as explained below it is possible to define a method for coding information, the method comprising obtaining a list of integers to code; determining a hyperspatial pyramid having a size suitable for encoding the integer list, the hyperspace pyramid having a plurality of vertices such that the degree of the hyperspace pyramid is equal to the sum of the integers of the integer list and the dimension of the Hyperspatial pyramid is equal to the integer number of the list of integers minus one; index the list of integers in this pyramid by means of an indexing system; and provide an indication of indexation in the hyperspace pyramid.

In one embodiment, the list of integers is obtained from a file. The person versed in the field will appreciate that various types of files can be used to store the list of integers.

It will also be appreciated that the step of providing indexing of the list of integers in the hyperspace pyramid may include in one embodiment indexing storage in a file. The person versed in the field will appreciate that various types of files can be used for this storage. Alternatively, it will be appreciated by the person skilled in the art that the step of providing indexing of the list of integers in the hyperspace pyramid may include in one embodiment transmitting the list of integers to a processing unit by through a network. The person versed in the field will appreciate that various types of networks can be used. In fact, the network may include at least one of a local area network, a MAN type network and a WAN type network. In one embodiment, the transmission will take place over the Internet.

It will also be appreciated that in one embodiment, an indication of the indexation of the integer list in the hyperspace pyramid will include displaying said indication of indexing the integer list on a screen. It will be appreciated by the person skilled in the art that a method is described for decoding information encoded by the method for encoding information described above. In this case, the method will include the steps of obtaining the indexing indication of the integer list in the hyperspace pyramid. The method will further include the step of obtaining an indication of an indexing system used to generate said indication of the indexation of the integer list in the hyperspace pyramid. The method will further include the step of determining by means of indexing and indicating an indexing system used said list of integers, and finally, the step of providing said list of integers. The diagram of Figure 4 illustrates an embodiment for providing several coding and decoding options for a list of integers <>.

In particular the function CODE is a family of encoding functions which index the list of integers <> with respect to the lattice T _mn where m is the total degree of the list and n - 1 its length. The result is a triplet <X, m, n>, X, representing the integer corresponding to the index of the projective point having the initial list of integers in input as vector of homogeneous coordinates.

The person versed in the field will appreciate that the calculated index is different according to the coding method adopted. Depending on the case, this triplet can be provided to the DECODE function that will restore the initial list or separated into two parts to form a single integer Z that can be decoded bijectively without any other information.

The left-hand part of the diagram of FIG. 4 shows another way of constituting a single integer W, allowing a bijective decoding. It is a family of functions denoted K ₂ , which couple the triplet two by two, m and n producing an integer, itself a pair at X. Successively applied two-dimensional decoding functions then make it possible to restore the initial code.

The right part of the diagram of Figure 4 corresponds to a bijective encoding-decoding scheme coupled to a combinatorial compression mechanism. In in this case, the triplet <X, m, n> is first transformed into a quadruplet, the integer X being replaced by particular combinatorial coordinates mx and nx. The quadruplet (mx, nx, m, ri) is then encoded by a four-dimensional coding scheme to produce a C code. This type of bijective code can be decoded by a four-dimensional inverse decoding scheme. The compression principle lies in the significant decrease in the degree of the quadruplet compared to the degree of the triplet, which largely offsets the increase of an additional character between the triplet and the quadruplet.

The indexing of a lattice makes it possible to code with a single integer the degree and the dimension of the lattice. The indexation of a lattice T _m "can be carried out by counting the number of squares of the Pascal triangle preceding the coordinate box x = m and y = m + n corresponding to the binomial coefficient. The number of squares is thus calculated.

triangle preceding the m, line of the lattice to which is added the degree of the lattice corresponding to its column. The formula found corresponds to Cantor's indexation of natural numbers in the Cartesian plane.

(m + n) * (m + n + \)

 I = - - - + m

 2

Global and total indexation

Global indexing makes it possible to ensure the bijection between the points of a lattice and their indexing in a n-dimensional space. Instead of calculating the number of cells preceding the binomial coefficient of the lattice, the total number of points of all previous lattices is calculated. From the equation T | 2 ^r , and noting that

2 + · · · + 2 ^r = 2 ^{r + 1} , the formulation of the global index I _g is obtained by summing the number of points of the lattices of the lines going from 0 to m + n - 1 1 and points of the lattices of the line m + n ranging from 0 to m: m-1

I _g = 2 ^{m + n} + ΣCm (i, m + n - \).

 = 0

The total number plus one, of points preceding the lattice Τ ₅ will be obtained as follows:

7 (5.2) → (128-I) + (1 + 7 + 21 + 35 + 35) + (1) = 227.

The global index added to the local index, the number of the point on the trellis, gives the total index of the point which makes it possible to find the initial coded vector without memorizing the degree and the dimension of the trellis. It should be noted that the bijective character of coding and decoding has a price to pay in terms of size of the generated indices and calculation time. On this last point, optimized versions of the following GLOBAL and LOCAL algorithms will be presented later as well as several methods for calculating the local index.

It will be appreciated by the person versed in the field that in the GLOBAL algorithm illustrated in FIG. 5, the variable ind is equivalent to the total number of points of the trellises preceding the trellis line T _{m + nm} and the variable acc equivalent to the summation of the binomial coefficients on the lattice line. The LOCAL decoding function illustrated in FIG. 5 performs the reverse search by locating on Pascal's triangle the line preceding the lattice and then isolating the lattice on its line.

The different recursive formulations of the binomial coefficients make it possible to consider the recursive trees of call of the functions as systems of ordering of the points of a lattice. In any case, the conditions for stopping the

recursivity will be either l, k = r, or = 1, k = 0, and in this case, the number of terminal leaves all equal to one will necessarily be equal to the desired binomial coefficient. FIG. 6 illustrates the recursive calls of the functions corresponding to the equations with r≥ \

for the calculation of the binomial coefficient The binomial coefficients corresponding to each node of the tree provides the number of terminal leaves of their respective subtrees. The different algorithms for reading and traversing trees thus provide algorithms for indexing the terminal leaves. By establishing a correspondence between the production of the terminal leaves and points of the lattice, it is possible to obtain a scheduling of the latter. In particular, deep tree courses consist of treating the root of the tree and recursively traversing the left and right subtrees of the root. In the prefix or lexicographic depth course, the root is processed before recursive calls on the left subtrees, then right.

Positional Projective Coding The representation of numbers in positional form is used to encode the sequence of coordinates of a homogeneous point. Let N be this number, a, the symbol and b the numbering base, by definition: n

 N = ¾> 'with0 a, <b -l.

The numbering base m + 1 is used where m is the degree of the lattice of dimension n, and a, is the homogeneous coordinate of rank.

The positional index I _p of the coordinate point [x ₀ : ...: x _n ] of the lattice T _mn is:

Figure 7 shows the example of the indexation of the 2-dimensional cubic curve of Figure 2. The indices are calculated in quaternary base. For example, the point [2: 0: 1] has

2. (4) ² + 0.Ç4) ¹ + 1. (4)

for index 1 1. The maximum index of the trellis being equal to 16 for

 3

a total number of 10 points, the figure shows a hollow indexation ie with holes in the numbering. The person skilled in the art will appreciate that the hollow indexing phenomenon is one of the disadvantages of positional coding. Indeed, for a lattice T _mnt the growth of the ratio p _mn between the maximum index (m + 1) "of the positional coding and the corresponding maximum index of a combinatorial coding is:

Figure 8 illustrates the representative curves of p as a function of the growth of m and n. These make it possible to visualize the expansion of the number of digits necessary for the

28531167061 1 positional coding. For example, a value of p _{lQ 10} equal to

 184756

The algebraic ring structure of the polynomials immediately makes it possible to show the additive homomorphism of the positional coding with I _p (A + B) = I _p (A) + I _p (B). In fact, let the points A of coordinates [a _Q : ...: _n ] and B of coordinates [b _Q : ...: b _n ], thanks to the properties of associativity and distributivity, it is then possible to verify the relationship :

/ (A + B) = Σa, (m + iy ₊ = ± -Σ (a + b) _t (m + iy These properties of additive homomorphism extend in the same way to linear combinations of type A + λ ^ Β The coding of the point 1 1 of Figure 7 from the linear combination of points 1 and 16, with respective weights of 1 and 2 will thus give:

1.1 + 2.16 "

..10: 0: 3] + 2. [3: 0: 0]

1 + 2 ^J Coding in numerical order

The definition of numerical order suggests a simple algorithm for calculating the index of a projective point, based on the column-by-column count of the coordinate position of that point. The left-most and non-zero difference of the exponent vectors of two projective points, where vm _a -ν? Η _β is negative, it suffices to count from left to right, column by column, the index of each coordinate and d summon it to find the final index of the point. To calculate, for example, the index of the point [1: 2: 1: 0] of the lattice ₄₃ of FIG. 11, 15 zeros before the 1 of the first column will be counted, 4 zeros and 3 ones before 2 of the second column and 1 zero before the 1 in the third column. The index 23 or 11 = (34 - 23) will be obtained according to the indexing convention chosen, ascending or descending.

It will be appreciated that the lattices T _mn are recursively decomposed into sub-lattices T _mn, T _m _ _{X n} _ _x , · * *, ^ o _{, «} -i ' ^this amounts to breaking down the representative polynomial of a hypersurface into polynomials hypersurfaces of decreasing degrees in the lower dimension.

Moreover, the exponent vectors are grouped in m + 1 tables corresponding to the monomials of the polynomials of the hypersurfaces of decreasing degree. Figure 11 shows the breakdown by arrays of a lattice (4.3). The last three columns of exponents represent lattice exponent vectors (4,2), (3,2), (2,2), (1,2) and (0,2).

Finally, the generated arrays are in turn decomposed into lower-order arrays in dimension n-2 and so on. The last two columns of the exponent vectors in the figure therefore represent the monomials of homogeneous polynomials in dimension 1. n i

h ⁼ ΣCm (m - 1 -Σvn [j], n + \ - i).

The NUMORD algorithm, shown in Figure 10, corresponds to the previous equation contracted. The second loop allowing the iterative accumulation of the degree of the vector of exponents is indeed absorbed inside the first loop by the instruction 4. In order to lighten the presentation of the code, the projective dimension equal to the length of the vector minus one is provided as an argument. Figure 1 1 allows through an example to visualize the step by step of the function NUMORD.

NUMORD {[\, \, \ M5) → {2 \ Q} _num

The index {210} ^ _um is the result of the summation of the pyramidal numbers carried out at lines 1 to 6 of the algorithm of FIG.

<)

This sequence shows the notion of pyramidal composition of the numerical index, illustrated from the table and the Pascal triangle of Figure 11. The decoding of the calculated indices, presented further on, will therefore be essentially based on the principle of decomposition. pyramidal of the proposed number, ie the search for the sequence in cascade of the pyramidal numbers composing it.

The examples of indexing which follow for points belonging to lattices T _{4 3,} T _{6 4, 55 9 ι} Γ ₅₄₉ illustrate in particular the non bijective character of the numerical coding

NUMORD ([\, 2, l, 0l3) → {U} " _um

NUMORD ([4.0, \, 0, l] A) → {U} _num

N ⁽ 7 O / Z ⁾ ([1,2,3,4,5,6,7,8,9,10], 9) → {22724037001} _num / ΟΛΖ ⁾ ([0,2,3,4 , 5,6,7,8,9,10], 9) → {22724037001} ix [i] ind m

 1 1 84 6 5

 2 0 14 5 4

 3 2 4 5 3

 4 1 1 3 2

 5 1 0 2 1

Decoding in numerical order

The decoding algorithm is based on inverting the indexing process between lines 1 to 6 of the NUMORD function. The objective for calculating the numerical signature vm of the exponents is to first find the successive pyramidal numbers which made it possible to calculate the numerical index. To find the largest pyramidal number in the composition of 7 ^m ™, look for the largest pyramidal number is less than or equal to it in column m of Pascal's triangle. Once this number has been found and its rank in the column, the first exponent of the desired signature is obtained and this is done from column to column by decrementing each time the starting index. These steps are illustrated by the preceding table, with the calculation of the digital signature of {210} ₅ . The last exponent of the signature is calculated simply by subtracting the sum of the exponents found from the degree of the index.

The following examples illustrate the enumeration of the lattice points T _{4 3} of Figure 11 indexed from 0 to 34 = Cr (4.3) -1, and then the enumeration of the lattice point Γ _{3 2} of FIG 2 indexed from 0 at 9 = Cr (3,2) - 1.

ORDNUM ([34- £, 4.3]) → [0.0.0.4], [0.0, 1, 3], · · · [4.0.0.0], k = 0, · · · 34

ORDNUM ([9 - £, 3,2]) → [0,0,3], [0, 1, 2], · · · [3,0,0], k = 0, · · ·, 9

Coding in lexicographic order In order to carry out a coding in lexicographic order, the lattice T _mn is decomposed into sub-lattices T _m _ _f _ T _m _ _{X n} _ _{2 i} ..., T _m _ _{l 0} , which amounts to to break down the polynomial representative of a hypersurface in hypersurfaces polynomials of a lower degree in smaller dimensions.

Then, the vectors vm of indices are grouped in n tables corresponding to the monomials of the polynomials of the hypersurfaces of decreasing degree. Figure 13 shows the decomposition by arrays of a lattice ₄₃ . The 4 tables correspond respectively to the lattices _33> T _{i2i 31} and ₃₀ , cubic in the dimensions

A = 3, ···, ().

Finally, generated arrays are in turn decomposed into lower-order arrays in decreasing dimensions. m-l

/, = C _n (m - i, n -1 -vm [i + 1])

Figure 13 illustrates the block decomposition of a lattice (4, 3). The signatures of length 4 beginning either with χο, χχ, x ₂ , or x ₃ form four blocks of respective sizes

= 1. The signature

vm = (1,1,2,3) has for subscript:

C (4, (3-I) -1) + C (3, (3-I) -1) + C (2, (3-1) -2) + C (1, (3-1) -3 ) = {10} _to .

The sequence of the function Z L LYO D D ([0, 1, 1, 2], 4,3) -> {23} _lex is shown in the following table:

The LEXORD algorithm is described in Figure 14. Decoding in lexicographic order

The ORDLEX algorithm for decoding in lexicographic order is described in Figure 15.

The following examples illustrate the regeneration of the points of the lattice T _{4 3} of FIG. 13 indexes from 0 to 34 = 0 (4.3) -1 and then the enumeration of the points of the lattice T _{3 2} of FIG. 2 indexes from 0 to 9 = 0 (3,2) - 1.

ORDLEX ([34 - k, 4.3]) → [0,0,0,0], [0,0,0,1], · · · [3,3,3,3], * = 0, · · ·, 34. ORDLEX ([9 - k,, 2]) → [0,0,0], [0,0, 1], · · · [2,2,2], k = 0, · · ·, 9. Projective gray coding

The summation of the variations between the consecutive codes produced for the digital coding is not constant. For example, the summation of the variations between the codes 34 and 33 of the lattice Γ _{4 3} is equal to (0,0,0,4) - (0,0,1,3) → 0 and equal to (0,0, 4.0) - (0,1,0,3) - 1 between codes 30 and 29. Projective Gray coding eliminates these differences of variation and generally jumps between non-consecutive points of a mesh. These points actually have a distance on the lattice greater than one. The principle of Gray Projective coding presented is essentially a replacement in the recursive tree, subtrees of odd indexing, by their symmetrical. This replacement being recursive, the operation will be repeated for each sheet as many times as it will have successive parents having odd indexations. Therefore, subtrees with an even number of odd successive parents will not have to be modified, even-numbered symmetries being canceled.

It will be appreciated that in a projective Gray coding, the summation of the exponents of the vector Vn _M - Vn _i obtained from two consecutive points is zero and the Hamming distance between two consecutive vectors of exponents is equal to 2.

Figure 16 illustrates a Projective Gray (4.3) coding.

The algorithm shown in Figure 17 is a modified version of the algorithm shown in Figure 10. Line 4 tests the parity of each exponent of the input signature (function type Q). In the case of an odd exponent, the indicium is inverted on line 7 and the variable s changes sign. Its value equal to 1 or -1 makes it possible to take into account the even or odd number of the exponents already treated and thus to identify the subtrees having an even number of odd successive parents who do not then have to be modified.

FIG. 18 illustrates the path of the points of the lattice T _{4 J} in projective Gray. The different sections show a succession of boustrophedonic Cantorian courses.

The projective gray doting of the point 10 for the lattice Γ ₆₄ and for the lattice T _{2 4} is obtained in the following way:

## STR2 ## ([4.0, 1.0, 1], 4) → (10);

GRAORD ([0,0, \, 0, l], 4) → {10} Projective gray decoding

An implementation of the Projective Gray decoding algorithm is illustrated in Figure 19 by the ORDGRA function.

The following example illustrates the enumeration of lattice points Γ _{4 3} in Figure 16 indexed from O to 34 = Cr (4.3) - 1:

ORDGRA ([34- £, 4.3]) → [0.0.0.4], · · - [3.0.0, 1], [4.0.0.0], k = 0, · · ^■ , 34. Dual and direct transcoding

It will be appreciated that transcoding from one coding system to another may be performed by going back and forth from one coding to the other by simple conversion between the exponent vector Vn and the vector of coordinates Vm, or by a direct equivalence between the indexing functions. In the first case, we will speak of dual transcoding performed by the two functions VN2VM and VMIVN and we will have the following equivalences formed from the lexicographic and numerical signatures:

S _num [Cm (m, n) - i] = Vm Vn (S _lex [i - 1]); S _lex [Cm (m, n) - i] = VnVm {S _mm [i -1]).

For example if:

S _num [Cm (7,4) - 68] = [1, 1,4,1,0] and 5 _ta [168-l] = [0,1,2,2,2,2,3] it will be obtained: [1, 1, 4, 1, 0] <- VM2VN ([0, 1, 2,2,2,2,3]) Transcoding between vectors is shown in Figure 20. Recursive production of a order

Given the symmetry relation of Pascal's triangle ( ^r and the equivalences between recursive decompositions by column and diagonal, a tree recursive is associated with two symmetric binomial coefficients. Figure 21 shows the recursive structure of. The nodes a (i, j) are

indexed in width Î and in depth j and the leaf numbers of the corresponding subtrees are directly calculated from i and j. Any function / (, j) then makes it possible to label the nodes of the tree and produces a certain type of scheduling of the terminal leaves of the tree. If one opts for the same lexicographic route prefix of the tree, one obtains the indexation in numerical order of a quartic curve in two dimensions in lexicographic order of a hyperquadric in or finally the indexation in order

lexicographic of 4-combinations without repetition for 6 objects. The recursive production of an order therefore amounts to associating with a binomial coefficient a triplet composed of a reading order of the recursive tree, a function of indexing the nodes of this tree and a choice of the type combinations with or without repetitions.

Indexing Functions of Recursive Trees The numeric, numeric and combinatorial indexing functions can be deduced sequentially from one another. The resulting geometric interpretation uses the passages between projective, affine and Euclidean spaces.

Numeric and numeric orders

The numerical indexing algorithm consists in incrementing from scratch the indexes of nodes of the same level as long as they have the same parent. The path of the tree is from left to right and from bottom to top. The index of the last level is obtained by deducing the sum of the indices of the previous levels of the degree of the tree, which guarantees the homogeneity of the digital points produced. To pass from the numerical coding space to the numeric coding space, passing matrices N2X _n which associate at the vertices of simplexes of digital projective spaces the vertices of projective subspaces with increasing co-ordinates. For example, the top [1: 0: 0: 0] projective simplex in 3 dimensions will be associated with the vertex [1,1,1,1]. The matrix product of a n-dimensional projective point with the matrix N2X _n _ _x will in all cases generate a point with increasing coordinates. By noting X2N _n , the inverse gate matrix allowing to go from the numlex to the numeric space, the following transition matrices will be obtained.

1 1 1 1 0

 1 1 0 1

 N2X 0 1 X2N = 0 0

0 0 0 1 0 0 0 1

As an example, the projective numerical point [ ₀ will be transformed into a point numlex [cc _Q , ₀ + _λ , α _ϋ + _λ + ₂ , ₀ + or, + a ₂ + a ₃ ] by the matrix product with matrix N2X ^. The point [5: 2: 0: 4] will be transformed into a numlex point [5: 7: 7: 1 1 and its lexicographic index will be calculated from its coordinates as

- follows: = 1001 + 84 + 28 + 5 = 1 1 18

Numerical and combinatorial orders

We move from numerical to combinatorial order by performing a linear combination between the numlex points and the point [0, 1, 2, ^■ · ·, «- 2, M - 1]. This operation amounts to adding each coordinate of the numlex point with its rank, equivalent to its level in the recursive tree. Figure 24 illustrates the presence of the combinatorial subspace of the 3-combinations in the reference simplex of the 3-dimensional projective space. This subspace is represented by the vertices of its reference simplex obtained by projective linear transformation of the vertices of the reference simplex of the initial projective space (the simplex unit). The following linear transformation is obtained:

The projective points belonging to the combinatorial space and obtained by any combination of two of the vertices of the simplex therefore have strictly increasing coordinates. It is possible to establish the following equivalence theorem: The indexation of a k-combination of an r-alphabet {0, · · ·, Γ - 1} is equivalent by projective transformation to the numerical indexation of the trellis T _r _ _kk .

FIG. 25 illustrates the numerical indexing of a lattice T _2Ji equivalent to the indexation of

3-combinations from 5-alphabet 0, · · ·, 4. In higher dimension, by

example, the 2172 digital point coordinated [2,4,1,2,0,3] T _l2 mesh belonging to ₅ is converted into dot numlex [2,6,7,9,9,12] by matrix multiplication with the matrix N2X ₅ , then is added with the point [0,1,2,3,4] to give the combinatorial point [2,7,9,12,13,17]. We then check that the position of the combination [2,7,9,12,13] for the alphabet <0,1, · · ·, 17 - 1> is:

2172

The equivalence between numerical and combinatorial order makes it possible to use ORDNUM numerical coding algorithms as fast algorithms for generations of combinations. These involve in particular enumerative algorithms type HAKMEM _m (R. Schroeppel, Mr. Beeler, RW Gosper. Hakmem. Technical report, Massachusetts Institute of Technology, 1972), generating combinations from the previous combination represented in binary form. As this type of algorithm, to generate the combinations, increasingly lists the numbers having the same number of bits 1 in their binary representation, it follows that the algorithm Optimized ORDNUM can also be used for this purpose. Figure 26 illustrates the enumerative generation of the projective points of a lattice Γ _{2 3} corresponding to a quadric in 3 dimensions. The 10 points, by projective transformation in the combinatorial subspace, form the 3-combinations taken from 5 starting from the 5 -alphabet 0, ..., 4. The relation with the binary representation of the combinations calculated by the HAKMEM algorithm follows with a transposition of the indices of the combinations enumerated decreasingly in the direction of the reading.

Numeric and Cantorian Orders

It is possible, using the previous results, to transcode the Cantorian coding from the numerical coding by making projective space changes to Euclidean. Using the matrix formalism, the passage of a numerical point [ ₀ : _x : ...: a _n ] from the projective space to the Euclidean space is in fact realized by the product of this point by the transit matrix P2E where \ a \ is the degree of the point. The matrix E2P will allow the inverse operation to move from the Euclidean space to the projective space:

The result is the following homogeneous point:

(a ₀ + ûf, + · · · + «")

 \ at \

The Euclidean point obtained by eliminating the last coordinate equal to 1, [α _ϋ : a _x : ^■■■ : _n _ _x ] is then the Cantorian point in n - 1 dimensions. An equivalence is thus established between the Cantorian coding of the inverse list K (x _k , ..., x _x ) and the numerical coding. To do so, it will be noted that the recursive Cantorian tree is obtained by eliminating the last level of the digital recursive tree, ie the level allowing the homogenization of the numerical point. By repeating the example

10 + 4 -1

it is possible to check that /, = 782. It should be noted that the first term of the equation:

£ (1,3,2,4) →

must be subtracted to get an indexation of the lattice points between 0 and the total number of points.

The equation allows on the other hand to obtain the second

equivalence, K (x _n , ..., x ₂ ) - Ι _ηνη (χ ..., χ _η ) which matches the Cantor encoding of a list, the numerical coding of the inverse list preceded by an integer any. This equivalence is illustrated by the following example which makes it possible to determine the coding of Cantor K ₃ from the function NUMORD ([xO, x \, x2, x], 3) _{used in} symbolic _mode :

_ m _Q , m, {m + 1) m ₂ (m ₂ + 1) (m ₂ + 2)

with m ₀ = _x 3, m _l = m ₀ + _{x 2} , m ₂ = m _x + x \. Taxonomy and Hamiltonian Order

The above equivalences make it possible to establish a taxonomy of the coding systems corresponding to different indexing orders. Figure 28 presents the numerical and combinatorial orders as special cases of numerical order, the Cantorian order being itself a special case of the combinatorial order. It is possible to extend this classification by introducing a Hamiltonian order which will also include the projective Gray order, itself considered as a generalization of the Burstrophedonic cantorian order. New orders based on spiral curves filling the space will be described below. Search for Hamiltonian chains on lattices

The person versed in the field will appreciate that the local indication for a trellis is dependent on the type of course of all of its points. Among the available methods are in particular those related to space filling curve theory. Cartesian spiral coupling function

The spiral coupling function shown in FIG. 29 makes it possible to index the relative integers two by two. The Cartesian plane is traversed in a spiral according to FIG. 29. In this case the indexing polynomials of the coordinate pairs of the Cartesian plane are searched. It will be appreciated that the generation of the spiral is performed iteratively using an isosceles rectangle square rotated by π / 2, then scaled and then translated by one unit so as to ensure the continuity of the curve.

4x - x + y if x> 0 and \ y \ <= \ x

4y ² - y - x if y> 0 and \ x \ <= \ y

4x ² - x - y if x <0 and \ y \ <= | x

4y ² - 3y + x if y <0 and | x | <= | there The point 44 has the entire square root 6, which corresponds to the sixth square of the spiral having the point 36 = 6.6 as the vertex. Point 44, is located between the vertex 36 (even square) and the next vertex 49 (odd square). Being higher than the intermediate angular point 42 = 7.6, it is therefore located on the lower horizontal section of the spiral. The variables / and j are then calculated from the intermediate angular point 42, where / ^' is thus calculated from the index formula of the lower left corner of the spiral I _B = 2 / (2 / + 1) = 42 i = 3 and j from the difference between the given number and the lower left index, ie j = 44 - 42 = 2. The computation of the pair [x, y] is completed by carrying the values of and j in the preceding formulas that is to say j = x + i then i = -y. The final result is [x, y] = [j -, - /] = [-1, -3]. The following example illustrates the use of the two spiral Cartesian coding and decoding functions SPIORDc and ORDSPIc.

SPIORDc ([- \ 509123456789,82739987654321]) →

{27383622228148684752295686632} _s e O ^ DS / c (27383622228148684752295686632) → [-1509123456789.82739987654321]

Figure 32 depicts an implementation of a Cartesian coding algorithm SPIORDc and Cartesian spiral decoding ORDSPIc.

Affine function of spiral coupling y (9y - 7)

 + x if y <0 and x <= 2y

 2

 x (9x- 7)

 if x> 0 and y <= 2x

 Z (x, y) = + y

 2

(9 (x - y) ² - x - y)

 other

Figure 33 depicts an implementation of an SPIORDa spiral affine encoding algorithm and ORDSPIa spiral affine decoding. Projective function of spiral coupling

It is possible to construct a spiral order in "-dimensions by recursively defining chains of spiral chains, the first spirals being one-dimensional, then two-dimensional, and so on. Figure 34 shows a spiral order for a lattice (6,4) of sexics in 3 dimensions. The junctions between the stacked two-dimensional spirals are alternated, sometimes between the beginnings, sometimes between the ends of the spirals. To obtain spiral chains in 4 dimensions, it is necessary to chain spiral meshes in 3 dimensions to each other starting with a point, lattice (0,4) then a lattice (1,4), a lattice (2,4) etc. Optimizations of the computation of the binomial coefficients

The person skilled in the art will appreciate that the rapid calculation of binomial coefficients is essential in all the coding and decoding functions presented above. Iterative or recursive functions based on the simple relations attached to the Pascal triangle are excellent candidates for the quick calculation of small binomial coefficients. They are no longer sufficient for calculations involving large integers and must be improved or replaced. Also variants and algorithmic improvements needed to reduce computing time will be presented below.

Recursive formulation

The recursive formulation with r≥ 1 of additive type, is the basis

of the construction of Pascal's triangle. In this case, the associated binary recursive tree has many redundant branches and on the other hand has a maximum depth of r -1. These properties give the computation algorithm a temporal complexity in the order of the binomial coefficient itself. As a result the calculation times

become immediately prohibitive and can only be controlled by computer using call tables that memorize the functions called, their arguments and their results. This mechanism borrowed from optimization theory of the compilation is particularly in the Maple language _had tp _e t break the computational complexity by eliminating redundant calls present in the recursive calculation of binomial coefficients also in that of the Fibonacci numbers. By moving to an additive-multiplicative formulation generalizing the formula, a new recursive formulation of depth

maximum r - k minimizing redundancy will be presented. It will be shown that this approach can be considered as an optimized recursive variant of the Vandermonde convolution. By applying the formula

repetitively on the binomial coefficients of an isosceles right triangle, the following relation is obtained:

This is obtained by noting, as illustrated in Figure 34, that the binomial coefficient at the lower point of the triangle can be calculated from the binomial coefficients of any of the lines above. All that is required is to assign the binomial coefficients of this line of an appropriate multiplicative factor, for example a0 = l .t / 0 + 3.c / l + 3.i / 2 + l .d3. These multiplicative factors due to the redundancies of the binary recursive formula 8 applied line by line and the binomial coefficient after binomial coefficient are simply the binomial coefficients of the (x + y) ^{d development} . Figure 34 shows the binary application tree of the formula with r≥l on a four-sided triangle. Duplications

successive binomial coefficients provide the following multiplicative factor suites {1}, {1,1}, {1, 2,1}, {1,3,3,1} and the following equalities α0 = 1.α0, α0 = 1.Ζ> 0 + 1.Μ, a0 = l .c0 + 2.cl + l .c2, a0 = l .uf0 + 3.c / l + 3. ^ 2 + 1. ^ 3. By posing s = k in the formula the intermediate recursive form

following is obtained:

In order to guarantee the existence of a right triangle associated with each binomial coefficient of Pascal's triangle, the leftmost symmetric coefficient on a line must be preserved using the symmetry property of the binomial coefficients

The following equation is then obtained for k≤r:

(A r - k

 \ k) i = 0 min (k - i, r - k ~ (k -)) min (i, k - i)

The latter can finally be considered as an optimized recursive variant of the Vandermonde convolution which is expressed by:

The algorithm of Figure 37 uses three stopping conditions according to the values of k. They have been disembarked for easier reading of the code. From the point of view of purely theoretical programming, the algorithm also works without resorting to the function min (), useless in the classic formulation of Vandermonde, but useful in the reduction of recursive calls.

By performing the following successive substitutions, k

 y {9y - l)

 + x if y <0 and x <= 2y

2

 x (9x - 7)

 + y if x> 0 and y <= 2x the equivalence between the relations Z (x, y) = and

 (9 (x - yy - x ~ y)

other and ions

possible among the lines of Pascal's triangle, the recursion starting from the pair of lines {r - k, k} is {6.3}. To be able to use in the convolution of Vandermonde, the intermediate lines between the lines r - k and k, it would be necessary to choose a r - 2k

couple {r - k - i, k + i} satisfying - ->/> 0 and consequently introduce an integer division by 2 in the recursion algorithm. In this example, there will be = 1 and the possibility to start recursion in the modified algorithm with the pair of lines {5,4}. Iterative Formulations There are three iterative formulations for calculating binomial coefficients from additions, subtractions, multiplications and divisions. It will be appreciated by the person versed in the field that the formulation by column is found in all the basic algorithmic implementations:

The line formulation is mainly used to calculate the expansion of Newton's binomials:

Finally the diagonal formulation comes from the property of symmetry of binomial coefficients with repetitions:

C _d (m, n) = C _y (n, m) = +) //.

/ = 1 Figure 38 illustrates an implementation of an algorithm for calculating binomials by XENUME line and YENUME column.

The binomial coefficient will be obtained in sixth iteration, by C (6,3) in the

column 3, at the end of the sequence YENUME (3,9) → <1,4,10,20,35,56,84> or else in the fourth iteration, by C (6,3) in line 9, to the end of the sequence XENUME (3,9) -> (1,9,36,84). The calculation time proportional to the iterations will generally make the fast calculation by means of the function C _x in the case where m> n and by means of C if m≤n. This calculation can be finally performed by the function C _d which requires the same number of iterations as C _x by construction of the Pascal triangle. The sequence to obtain the binomial coefficient 84 will then be

 (1,7,28,84). The generalization of this example leads us to use the property of symmetry to optimize the number of iterations of the computation of a binomial coefficient and to use a single function of computation in the two cases of figure:

\ C _y (m, n) if m>

\ C _y (n, m) If m≤n ^' This first optimization on the number of iterations being done, it is possible to reduce this number by half again by putting the equation C ^ (w, «) =] ^~ [(m + i) / i in the form of a product of differences of squares. Let the following product have an odd number 2k + 1 of terms,

(m + 1) (m + 2) (m + 2k) (m + 2k + 1),

 - 7. Γ7-: - after changing the variables m = M - k ~ l,

(M - k) (M - k + 1) M (M + k - 1) (M + k)

1 2 * + l 2k 2k + By grouping all the terms except for M in the form of the difference of two squares:

() ( ² - 1 ² ) ( ² -2 ² ) (M ² - k ² )

 1 2.3 4.5 2k. (2k + 1)

The same operation is obtained for a product having an even number of 2k terms and it is then obtained:

(M) (M - k) (M ² - 1 ² ) (M ² - 2 ² ) ( ² - (Å: - 1) ² )

1 1 2.3 4.5 ^" 2k (2k + 1)

By dividing at each iteration by 2 / (2 / + 1), the function of the optimized algorithm for calculating a binomial coefficient is then presented in FIG. 39.

Deferred divisions

Depending on the relative performance of multiplication and division, it is possible to consider variants of the previous algorithms by differing and limiting the exact divisions. To do this, it suffices to calculate the denominators independently which are factorials and to divide by this result at the very end of the algorithms or intermediately if one wants to control the expansion of large integers being computed. This approach is also interesting in the design of parallel algorithms.

Figure 39 illustrates an embodiment of a product calculation algorithm of differences of squares.

Dynamic programming calculation

The calculation of binomial coefficients can be done without multiplication and division

in the heart of the world + and using

storage memory. The necessary operations are the addition and the assignment and the number of passages required to compute a binomial coefficient is mn + ^ (m (m + 3)) + n + \. The principle is that a schema of computation of the binomial coefficients is established and is based on a rectangular rectangular representation of Pascal's triangle. The binomial coefficients are then evaluated column by column instead of line by line. The algorithm therefore consists of going through the boxes of a single rectangle as illustrated by FIG. 40. The binomial coefficient sought is located at the tip of the rectangle and calculated from the binomial coefficients of the preceding columns.

An implementation of the dynamic programming algorithm is shown in Figure 34.

Additive and multiplicative method Using the multiplication operator, it is possible from the rectangular representation of Pascal's triangle to develop methodologies for calculating binomial coefficients based on the rapid exponentiation of particular triangular matrices. It is for this appeal to the formula of the binomial theorem extended to switchable square matrices in this case (M + I) where ^r / is the identity matrix. The application of the formula makes it possible to express the powers of triangular matrices formed exclusively of 1, from matrices expressed from the binomial coefficients and it is then obtained:

Taking into account the symmetrical structure of the matrix M with respect to its diagonal, the matrix product of two matrices for any powers simplifies itself in the form of the function MATPRT (see figure 42) which uses for the calculations only two vectors VnO and Vn \ corresponding to the first columns of the two matrices, and stores the result in the vector Wn. Thus, for example, the matrix product of the 3 rd and 31 th columns of Pascal's triangle will be represented in rectangular form. The last number of the resulting vector Wn _will therefore be evaluated from the matrix product between two powers of the matrix M. In this case, the number 8436 _will therefore be evaluated from the powers 4 and 32 of the initial matrix having [1,1,1,1] for representative vector.

MATPRT {[\, A, 10,20], [1, 32,528,5984], 3) → [1, 36,666,8436]

The naive method therefore consists in calculating all the successive powers of M up to the desired power according to the following sequence:

MATPRT (\, \, \ M [\, \, \ M?>) → W _M ² - [1, 2,3,4]; Μ47 ΛΓ ([1,1,1,1], [1,2,3,4], 3) - ¾ = [1,3,6,10]; TPRT ([l, l, l, \, r, - - -, - - -], 3) → W ^ [l, \ + r, - - -, - - -].

To minimize the number of matrix products to obtain the binomial coefficient sought is to find the best decomposition of the corresponding power into a sequence of integers. The algorithmic strategy is then to calculate any binomial coefficient by a minimum number of matrix multiplications between the columns. Various methods for choosing columns can be used, some of which are borrowed from fast exponentiation techniques. These are based on the following relation applicable to integers but also to the square matrices: a ⁰ * = a ° a ^k .

The exponent ^e is decomposed into a sequence of integers by means of different numbering bases or other decomposition techniques such as Zeckendorf based on Fibonacci numbers. Figures 43a and 43b illustrate this principle for the matrix calculation of binomial coefficients with a binary decomposition of the exponent (Figure 43a) or a Zeckendorf decomposition (Figure 43b). For a binary decomposition of the power, only columns 2, 2, 2, · - ·, 2 will be used to perform the matrix products. The binomial coefficient 2555190

will be calculated according to the following steps:

- The binomial coefficient is first expressed in the Pascal triangle represented in rectangular form:

The index of the column counted from is then represented in binary form which makes it possible to show the powers of used in the binary decomposition:

86 + 1 = 87 → {10101 1 1} ₂ → {2 ⁶ , 0.2 ⁴ , 0.2 ² , 2 ¹ , 2 ⁰ };

The successive matrix products of the corresponding matrix powers are carried out:

M ⁸⁷ = M ^M x M ¹⁶ x ⁴ x M ² x ¹ to finally provide the matrix to the desired power showing the binomial coefficient sought as the last element of the first column of the matrix:

1 0 0 0 0

 87 1 0 0 0

^ 64 + 16 + 4 + 2 + 1 ₌ 87 ₌

 3828 87 1 0 0

 1 13564 3828 87 1 0

 2555190 1 13564 3828 87 1

The exercise can be repeated using this time a decomposition of Zeckendorf:

87 → {55,21, 8,3} _Zk with the gain of a matrix multiplication compared to the binary decomposition:

M ⁸⁷ = M ⁵⁵ x M ²ⁱ x ⁸ x ³ .

In both cases, the efficiency and the minimization of the calculations require a simple iterative scheme of calculation of the intermediate matrix powers. In general, each matrix used in the resulting matrix product will be obtained from the matrix or matrices calculated previously. An iterative scheme of type

M _t , x M, will allow to calculate the matrix product on the fly for a binary decomposition whereas an iterative scheme of type, = M, x ₍ _, _{will be} used for a Zeckendorf decomposition with powers equal to the successive numbers of Fibonacci.

An implementation of the simplified matrix product algorithm is illustrated in Figure 42.

Factor calculation

It will be appreciated by the person versed in the field that the criteria of divisibility of the binomial coefficients can be exploited to find fast factorizations making it possible to obtain the binomial coefficient by multiplication of the factors found. For the calculation of large binomial coefficients, the best-known algorithm is Goetgheluck's algorithm (Goetgheluck, P., Computational Binomial Coefficients, The American Mathematical Monthly, 94 (4): 360-365, 1987.), based on a theorem of Lucas concerning the factorization of binomial coefficients in prime numbers.

Position of a number in Pascal's triangle

The search for the position of a given number, ie the identification of the two adjacent binomial coefficients of this number in a given row or column, is at the basis of the decoding techniques presented. This problem is undetermined when the row or column of membership is not specified because there may be several identical entries in the Pascal triangle. In the case where column n is known, it amounts to looking for a certain value yo and y = 0 the positive root of the polynomial function and the following equation:

(m + \) (m + 2) - - - (m + n)

y = y ₀ ; ·

 or

In this form the graph of the function is translated to obtain the searched root on the y-axis. Figure 44 illustrates the search for the position of the number 20 in the fourth column of the Pascal triangle with a graph of the function this time without translation. The graphical result shows that the abscissa of the intersection of the ordinate line 20 with the curve is situated between the values 2 and 3. The number 20 is therefore

 (4 + 2) (4 + 3)

between the binomial coefficients = 15 and = 35.

V ⁴ JV ⁴ J Naive method The naive method is the

with a stop test when the value of the calculated binomial coefficient exceeds the input value y ₀ . A column is therefore traveled iteratively from its origin until the discovery of a binomial coefficient greater than the given number. If the theoretical complexity is linear, the estimation of the actual costs of calculation must take into account the operations of addition, multiplication and exact division for large integers.

Dropper method

Some types of Diophantine equations have the common characteristic of having a single root greater than one, and of being derivable continuously. A search algorithm of this root is proposed for any diophantine equation of form y = P (x) where P (x) is a polynomial with positive integer coefficients or nulls and which has a single positive root. Specific equations that meet these criteria include, but are not limited to, the following equations: P _m (x) = x (x + k. \) - - - (x + k (m - \), S _m (x) = \ ^m + 2 ^m + - - - + (x - \

P _m () when k = 1 is found in many problems related to the existence of exact powers equal to a sequence of consecutive integers and to the heart of the theorem of Erdos and Selfridge (P. Kirschenhofer, A. Pintér, RF Tichy Yu, F. Bilu, B. Brindza, and A. Schinzel, Diophantine equations and bernoulli polynomials, Compositio Mathematica, 131 (2): 173-188, 2002.). When k = Q, P _m (x) is reduced to the classical exponential equation and in search of the same integer roots, S _m (x) is relative to Bernouilli numbers.

Principles The principle is to improve the naïve method described above by first searching for the number of digits of the positive root of the Diophantine equation, then calculating one by one the missing digits. The search for the number of digits is done by calculating the sign of the function for successive powers of 10 of the variable x. The order of convergence therefore adapts automatically to the degree of the function. The missing digits require at most 10 times the number of digits of the root, evaluations of the function.

Figure 45 illustrates an implementation of an algorithm of the dropper method ENCDEC.

Noting that the index returned by ENCDEC {\, x) is 0, it is obtained for example ENCDEC (496.5) → 6, and in general it is verified that the position of the central binomial coefficient respects the identity :

ENCDEC (Comb (x, x), x) → x.

By replacing on the other hand the function Comb by the function NUMFCT (x, n, 0) (algorithm presented in FIG. 46) in the functions ENCDEC and DIGBIN, and considering NUMFCT (x, n, 0), it is possible to carry out the extraction of the root itself integer of x, using the generalization of the ENCDEC function in search of the solutions of some families of Diophantine equations.

Analytical methods

The analytical methods for "= 1,2,3,4 may be interesting, depending on the performance of square and whole cubic root extraction. This is the case for example in two dimensions for the inverse function of the coupling function of

Cantor + y which makes it possible to determine the coordinates of a point

 (m + 1) * (m + 2)

identified by its index. Given the equation y ₀ - - ~ - = 0, we first look for the positive real value of m ₀ = ^ ⁺ ^ ⁺ which indicates between which diagonals of the Cantor scheme the desired point is. The integer part of m ₀ which is equal to the index of the diagonal preceding the desired point, then makes it possible to establish the

⁽ χ + y) * (χ + y + 1)

following correspondences with the equation I _K ^ = ^■ - + y and to determine the coordinates of the given point: y = y _a (Lm ₀ SJ + 1) J_ * ± (_LmJ ₀ > _] + 2)> _.

The point of index 1 1 will thus give a value of Lw ₀ J = 3 for the coordinates [3.1]. The analytical formulas for the values "= 3.4 will be illustrated below in the binomial decomposition methods of a number.

Numerical method

The convergence of the calculation of the position of a point can be significantly accelerated by searching for the roots of the binomial function with an iterative scheme using the successive derivatives of the function. A Raphson-Newton type algorithm whose principles could be extended to Householder type algorithms with higher convergence orders is presented below. Whichever algorithm is chosen, it must be designed for integer calculation and adapt to a function of any degree. For this purpose, a recursive algorithm for calculating the derivative A -th of the function is determined which calculates the value of the derivative without having to resort to a precomputed symbolic derivation function.

The recursive formulation of the k-th derivative of the binomial function is

0 if n <0 or k <0

nlf ₍ ([' ^k ) ⁾ 1 if n = 1 and k - 1

 ./¾ (m + n) + kffcj otherwise

Figure 47 illustrates an implementation of a recursive calculation algorithm of the k-th derivative DRVREC.

! / (m) = (m + 1) (m + 2) - - - (m + 7)

DRVREC (7,3) → 2 \ 0m ⁴ + 3360m ³ + 19320m ² + 47040m + 40614

Figure 48 illustrates an implementation of a Raphson-Newton algorithm of degree n. DRVSYM (4) → x - ((x ₊ 1) (* ₊ 2) (x ₊ 3) (* ₊ 4) -24 ^ ₀ )

 (((2x + 3) (x + 3) + (x + 1) (x + 2)) (x + 4) + (x + 1) (x + 2) (x + 3))

Pyramidal representation of whole numbers

Equipped with calculation functions to position any integer in a given column of the Pascal triangle, it is possible to perform the general function of decoding an integer in Pascal representation. The decoding of a number in quaternary representation, being a frequent operation and which we will see essential in the process of compression of the general lists of integers, a decoding of purely analytical nature will be described first, using the presented principles. previously. In this case, a sequential decoding is obtained where the successive binomial coefficients β ₃ ... ₀ are calculated from the positive integer roots of

",. (m + 1) (m + 2) --- (m + n),,,. . . . ... equation y = y ₀ - - - for degrees 4, ···, 1 of the variable m.

 not\

 Consistent in two dimensions to the index of the diagonal preceding the point corresponding to the given number, these integer values represent the index of the hyperplane, plane, line and diagonal point preceding the point sought.

Using the previous example where the combinatorial code was 1118, the decoding will be carried out according to the following sequence:

= 1001

Α0>) = β (ΐιΐ8-ιοοΐ) = = 84

¾Ο = Α (1118-1001 = 28 /? oO) = A, (1 118 -1001 - 84-28)

The vector [(4 + 1), (6 + 1), (6 + 1), (10 + 1)] is then multiplied by the matrix X2N _} to restore the original code [5: 2: 0: 4].

If now the dropper method is used, the BINRP function, shown in Figure 49, will be defined which searches for the position of any integer in column 4, then decrements that number of the found index in order to find the position of this new number in column 3, this process being continued to the very first column. The clues found are equivalent to the 4-dimensional Cantorian affine coordinates. For the previous example, BINRPA \ 1 18) → {5,7,7,1 1} corresponding to the vector previously obtained by the analytical method. In the same way, the quaternary binomial representation of the number 100 will be {100} ^ _M = {0,4,4,5} and its quaternary decomposition will be either {100} "= {0,10,20,70}.

Figure 49 illustrates an implementation of a BINRPA quaternary combinatorial decomposition algorithm of a number. It will then be possible to extend the BINRPA function to the n-ary decomposition of a number to obtain the general function BINRPN and its inverse function RPNBIN which recompose a number, given in Pascalian form, in decimal number.

Figure 50 illustrates an implementation of an N-ary pyramidal representation algorithm (BINRPNi). For example BINRPNi (l25.5) → {4,4,4,4,4} or {125} ^ = {4,4,4,4,4}, and vice versa RPNBINi ([A, A, A, 4 , A], S) → 125.

On the other hand, given the equivalences demonstrated previously, it is possible to replace the lexicographic decoding functionality ORDLEXQ by: ORDLEX ([x, m, n]) ≡ [n, - - -, n] - BINRPNi (x, m).

Finally, it is possible to obtain the coding and decoding functions of the combinatorial numeration system of paragraph 4.4.1, replacing a by a + ind in the two lines of BINRPNi and Comb (vn [n - i + \] - \, i) by Comb (n -vn [m + \ -i] - i, i) in RPNBINi.

Algorithmic Optimizations

The person versed in the field will appreciate that the elimination of some loops is the first level of algorithmic optimization. The basic principle is to replace a double loop the call to compute functions of a binomial coefficient in a loop by a simple loop calculating the coefficient iteratively.

Application to general lists of integers

Principle of combinatorial compression

Different types of coding are presented which, according to the diagram disclosed above, make it possible to encode any list of integers by means of a single integer, the total index or by a triplet consisting of the local index. of the list, its total degree and its length. The total index corresponds to the absolute numbering of a projective point representing the list to be encoded in the hyperspace projective space. The person versed in the field will appreciate that it is algorithmically pure but very quickly generates very large integers. Following the demonstrated equivalences between the digital coding and the Cantorian coding, it is also possible to opt for another strategy and to encode the triplet {Χ, πι, η} into a single integer ensuring the bijective character of the decoding coding. It will be possible for this purpose to use for example a three-dimensional Cantorian coding, or a two-dimensional Cantorian coding in two successive iterations. Implicit knowledge of the chosen coding method will allow decoding. Another method is to replace the number X by its two-dimensional Cantorian coordinates m _x , n _x and then to code the quadruplet {m _x , n _x , m, n} with four-dimensional Cantorian coding. By replacing the triplet by a quadruplet there will be a limited expansion effect of the size of the generated code: the length of the list to be encoded is simply incremented by one, but a significant compression effect due to the lowering of the total degree of the list to code. Indeed,

 (γγΐ - - f \ Λ * (YYl Yl -f- 1 \

a total weight of the triplet of ^xx - - + m _x + m + n, expressing X according to its Cantorian coordinates, the total degree of the quadruplet will be lowered to m _x + η _χ + m + n.

Figure 51 illustrates an implementation of a global (GLOBALo) and local (LOCALo) coding optimization algorithm.

Figure 52 illustrates an implementation of a digital decoding algorithm (ORDNUMo and ORINUMo).

Figure 53 illustrates an implementation of a compressed coding and decoding algorithm (INDCOM).

Taking again the projective point [69,78,73,71,77,65] of degree 433 in dimension 5 of example 4.1, the following indices are obtained: / _Wum = {55014101437} _(433.5)

I _To , = {1419606883389857208104148062281258856159455782592418086487285545

274686109596480318996466895925319463985864300012238628776356963859010} _z and in compressed form,

I _Com = / N CO ([69,78,73,71,77,65], 5) → {507098669008328947049} _c . Representation of lists of lists

The recursive application of the INDCOM function makes it possible to code lists of lists. Let L ₀ , L _{ , ^■■ ·, Σ "be a set of integer lists of respective dimensions"",",, · · ·, "", a first-level encoding will be defined as follows: INDCOM ([INDCOM (L ₀ , n ₀ ), INDCOM {L _x , «,), · ^■ ·, INDCOM (L _n ,« ")], w). Representation of relative integers

In addition to the representation of relative integers by spiral coupling functions, it is possible to consider the theoretical representation of signed integers, in pairs. This approach leads for example to double the length of the initial vector by preceding each integer by an integer signifying its sign and by extension its type. It is possible for example to use the convention of 0 for a positive integer, 1 for a negative integer, 2 for an irrational, 3 for an imaginary, and so on. The list {-1,2,7, -3,5} will thus be encoded in interlaced form {1, 1, 0,2,0,7,1, 3,0,5} with a compressed index of {6592155939510} _c .

The person versed in the field will therefore appreciate that relative integers may be encoded by means of a list of integers. In such a case, a list of integers to be encoded will be generated by means of the list of integers.

Alphanumeric sorts The numerical order allows to create dictionaries by alphanumeric tris. For this, the principle will consist in bringing all the words of the dictionary to identical lengths and weights. The words to be sorted will be associated with projective points of the same lattice. It will be taken for length, that of the largest word, and for weight, that of the heaviest word. To sort, for example, the words ART, ARA, ARMS, ARABIC, ARABICA, coded with an alphabet of 26 letters, the index of each letter corresponding to its position in the alphabet, the following points that correspond to them on the lattice T ₄ o, 7 will be scheduled. The maximum length will be inherited from the word ARABICA and the maximum weight of the word BRAS:

ART → [1,18,20,0,0,0,0,1] → {45609855},;

 num ARA → [1,18,1, 0,0,0,0,20] → {45662979},;

 num

ARM - »[2,18,1,19,0,0,0,0] -> {38531317}, ARABIC → [1, 18,1, 2,5,0,0,13] → {45658898},

ARABICA → [1, 18,1,2,9,3,1, 5] → {45658530}

By inverting the sort, to respect the descending order of the numeric coding, the alphabetically sorted sequence will be obtained: (45662979,45658898,45658530,45609855,38531317) →

(ARA, ARAB, ARABICA, ART, ARM).

hash

It will be appreciated by the person skilled in the art that the indexing functions presented above may advantageously serve as hash functions in the same manner as the conventional polynomial hash function constructed from the equation:

H (x) = ( ₀ .B ^k + _x .B ^k'x + - + a _k .B °) mod N.

In this case, B will be taken commonly equal to a power of 2, for example 256 and N equal to a prime number. By coupling the combinatorial indexing functions to the modulo operator in the same way, the following values will be obtained from the previous sequence. The choice of N will cause a collision phenomenon and a fill rate of the hash table more or less important.

(45662979,45658898,45658530,45609855,38531317) → (11,12,8,5,6)

Application to polynomials Literal representation

Considering the polynomial in the form of a character string has certain advantages, particularly for preserving the initial representation of the polynomial, for example in factored form or while preserving its type of notation, in reversed polonaise, for example. In return, the subsequent algorithmic processing of the polynomial must go through a conventional mini-compilation stage of expression type arithmetic. A character-by-character elementary encoding can therefore be made from a reduced alphabet of type {0,1, 2,3,4,5,6,7,8,9, x, +, -, V, (,), #} This is made up of digits, arithmetic operators, parentheses, the variable x, and the pound sign used as a separator. The alphabet will be extended to the letters of the alphabet if the polynomial is for symbolic processing. A numerical code being assigned to each symbol of the alphabet, the equivalence table of the alphabet with n characters will simply be the list of n + 1 natural numbers. The maximum weight of this list will therefore be (n + \) ² . The following example presents the homogeneous equation of a radius torus a and b centered at the origin of the Euclidean coordinate system: Tore = (x ₀ ² + xf + x ₂ ² - (b ² - a ² ) * x ² f - 4 * a ² * (x ² + x, ² ) * x ² .

The list of 56 integers and of total degree 3789, representing the equation of the torus in the UNICODE coding system is as follows:

^ _"= {40,120,48,94,50,43,120,49,94,50,43,120,50,94,50,45,40,98,94,50, 45,97,94,50,41, 42, 120.51, 94.50, 41, 94, 50, 45, 52, 42, 97, 94, 50, 42, 40, 120, 48, 94,

50.43, 120.49, 94, 50.41, 42, 120.51, 94.50} _{(Αι (∞Λ)} .

The numerical coding of L _Tare , is carried out by NUMORD (L _Tore , 56 - 1) which gives a local numerical index of:

I _Num = {430498654249915943678023780125509627688494209313620775647407 4479473688920429540259225190355106304329193124705002316203895665} _{(3789 55)} . Structured representation

Algorithmic performance problems are closely related to selected representations, and memory problems are partly caused by losses due to hollow polynomials. The vector L representing the numerical expansion of the torus polynomial for rays a = 2 and b = 1 makes it possible to visualize the different problems related to the vector representation: the order of the coefficients corresponding to the order chosen on the monomials, the presence of the zero coefficients specific to the hollow polynomials and finally the management of the signs of the coefficients which escape the encodings of the natural numbers. L = [1,0,2,0,1,0,0,0,0,2,0,2,0,0,1,0,0,0,0,0,0,0,0,0 , 0, -10.0, -10,0,0,6,0, 0,0,9]

After coding the negative values of the vector L, the numerical code of the new interlaced vector of the torus is then:

I _Num = {191438651803617499325873813948513} _{(46 69).} In order to overcome this type of problem and considering that the null coefficients affect only the size of the vector to be encoded and not its degree, it is possible to adopt the following cascade addressing strategy. A hash table with no collisions of a size equal to the number of coefficients Cm {m, ri) is created for a polynomial of degree m in n dimensions. This table contains the addresses encoded in turn as integers on the information specific to the monomials. This separately stored information can therefore be the signed numerical values of the coefficients, but also pointers on symbolic calculation functions of the coefficients or relational database indices to establish geometrical constraints between different polynomials (variational CADCAM). T ₀ = [k ₀ , 0, k ₂ , 0, k,, 0,0,0,0, k ₉ , 0, *,,, 0,0, k _u , 0,0,0,0,0 , 0,0,0,0,0, k ₂₅ , 0, k ₂₁ , 0, 0, k ₃₀ , 0,0,0, k _M ] Vector coding and hollow polynomials

The encoding of the hollow vectors can be realized in several different ways. The most common method is the so-called rank-and-value method described by Lehmer (DH Lehmer, The Machine Tools of Combinatorics, in EF Beckenbach, Editor, Applied Combinatorial Mathematics, pp. 5-31, Wiley, New York. 1964). In this case, the coding is performed by a sequence of integer pairs formed of the position of each integer in the hollow list, and then of its value. A zero is put at the end of the list to signify the end of the message. The previous table To is thus coded as follows:

TQ ₀ - (0, k _Q , 2, k ₂ , 4, _ki , 9, k _<) , \, k, 14, k ^^, 25, k ₂₅ , 27, k, 30, k- _i0 , 34, k _i , 0). The positions of the non-zero values can also be encoded separately using, for example, the combinatorial compression technique. Then there will be the following compressed code to store the locations of the non-zero values of To:

(35,0,2,4,9,11,14,25,27,30,34) → {1836997097013699578176773581776} It is also possible to use the Pascal coding for the decreasing sequences but by separately storing the length of the vector:

(34,30,27,25,14,1 1,9,4,2,0) → {148139661} _¾¾ ^■

Symbolic product of multivariate polynomials

The fast indexing functions of monomials which make it possible to produce the product of two multivariable polynomials in a vectorial way by direct indexing must then be used. The MPLNUM algorithm illustrated in FIG. 54 uses indexing in numerical order by multiplying each monomial of the first polynomial by each polynomial of the second, which is known as the naive multiplication algorithm of two polynomials. One of the advantages of the algorithm presented is that it can be easily extended to the multiplications of hollow vectors.

The symbolic multiplication algorithm of two multivariate polynomials is based on the elementary operation of the product of two monomials, the indexation of the resulting monomial and the summation of the monomials of the same rank. The MPLNUM function realizes the symbolic product of two polynomials in a given dimension. The degree of the resulting polynomial dgr calculated in line 1 and the numbers of the monomials of the two multiplied polynomials and the resulting polynomial are calculated from line 2 to line 9. The initialization of all the terms of the resulting vector to zero is carried out at the line 10. A loop is made on the number Iga of the monomials of the polynomial a in line 1 1 and on the number Igb of the monomials of the polynomial b in line 12. The signatures in numerical order of the two polynomials a and b are obtained on lines 13 and 14. The multiplication of the monomials is carried out by the addition of the signatures siga and sigb to the line 15. The index of the resulting monomial is calculated in line 16. Finally, the resulting monomials of the same rank are summed and accumulated in the vector associated with the resulting polynomials at line 17.

The following example illustrates the two-dimensional multiplication of a projective second degree (conic) curve by a first degree curve (right), which gives a third degree curve (degenerate cubic). Let yla be the conic of equation yla ^~ ₀ χ ₀ ² + a _x x ₀ x _x + a ₂ x _x ² + Û ₃ X ₀ X ₂ + α χ _{ χ ₂ + a ₅ x ^ and ybl, the right of equation ybl = b ₀ x ₀ + b _x x _x + b ₂ x ₂ , the 10 = coefficients of cubic y3c in numerical order

(0,0,3), (0,1, 2), · · -, (3,0,0) are obtained,

MPLNUM ([α, α1a, α1, α4, α5], [δ1, bbl, b1], 2, 1, 2) → [aO * bO, aO * b1 + al * b0, al * bl + al * bO, α1 * bl, α0 * bl + α3 * b0, α1 * bl + α3 * bl + α4 * b0, α1 * bl + α4 * bl, α3 * bl + α5 * b0, α4 * bl + α5 * bl, a5 * bl].

Implementation of the disclosed method

It will be appreciated by the person skilled in the art that the method for encoding the information disclosed herein as well as the method for decoding encoded information using the method disclosed herein may be implemented in a variety of ways.

In particular, these methods can be implemented for example in a computer 300 as shown in FIG. 30. The computer 300 will comprise, for example, a processor 310, a data bus 312, a data storage unit 314, a port communication unit 316, a display unit 318 and a keyboard 320.

In this example, the processor 310 is adapted to process instructions. The person skilled in the art will appreciate that various types of processors can be used to perform the instruction processing.

The data storage unit 314 is adapted to store data. The person skilled in the art will appreciate that various types of data storage units may be used to perform such storage. The communication port 316 is used to exchange information between the computer 300 and another processing unit not shown in FIG. 30. The person skilled in the art will appreciate that the communication port 316 depends on the type of connection used. between the computer 300 and the other processing unit. The display unit 318 is used to display data to a user. The person skilled in the art will appreciate that various types of display units may be used.

The keyboard 320 is used to allow a user to provide data to the computer 300. Again, the person skilled in the art will appreciate that various types of keyboards may be used.

It will be appreciated that the processor 310, the data storage unit 314, the communication port 316, the display unit 318 and the keyboard 320 are operatively connected together by means of the data bus 312.

In fact and in one embodiment, the data storage unit 314 includes an operating system 322. The operating system 322 may be of different types.

The data storage unit 314 furthermore comprises a program 324 that can be executed by the processor 310 and whose execution makes it possible to encode information. It will be appreciated by the person skilled in the art that a program whose execution will decode information may also be stored in the data storage unit 314.

The program for encoding information includes instructions for obtaining a list of integers to code. It will be appreciated that the list of integers to be encoded can be obtained in various ways. In one embodiment, the list of integers will be entered by a user using the keyboard 320. In another embodiment the list of integers will be obtained from a file located in the data storage unit 314 or received by the user. intermediate of the communication port 316.

The program for encoding information further includes instructions for determining a hyperspace pyramid sized to code the list. of integers, the hyperspace pyramid having a plurality of vertices whose number is determined by the degree of the hyperspace pyramid which is equal to the sum of the integers of the integer list and by the dimension of the hyperspace pyramid which is equal to number of integers in the list of integers minus one. The program for encoding information includes instructions for indexing the list of integers in this pyramid by means of an indexing system. As mentioned above, various indexing systems may be used.

The program for encoding information includes instructions for providing an indication of the indexing of the integer list in the hyperspace pyramid to thereby encode said list of integers to be encoded. It will be appreciated that the indication of the indexing of the list of integers in the hyperspace pyramid can be provided in various ways. For example, the indication of the indexing of the list of integers may be displayed on the display unit 318 (for example a screen). Alternatively, the indication of the indexing of the list of integers may be stored in a file located in the data storage unit 314 or it may be transmitted via the communication port 316 to another unit of data. treatment.

It will be appreciated that an information storage device may be provided for storing executable instructions by a processor. This information executable by a processor will cause the processor to execute a method including obtaining a list of integers to code; determining a hyperspatial pyramid having a size suitable for encoding the list of integers, the hyperspace pyramid having a plurality of vertices the number of which is determined by the degree of the hyperspace pyramid which is equal to the sum of the integers of the integer list and by the dimension of the hyperspatial pyramid which is equal to the number of integers in the list of integers minus one; index the list of integers in this pyramid by means of an indexing system; and providing an indication of indexing the list of integers in the hyperspace pyramid to thereby encode said list of integers to be encoded.

Clause 1. A method for encoding information, the method comprising:

obtain a list of integers to code; determining a hyperspatial pyramid having a size suitable for encoding the list of integers, the hyperspace pyramid having a plurality of vertices the number of which is determined by the degree of the hyperspace pyramid which is equal to the sum of the integers of the integer list and by the dimension of the hyperspatial pyramid which is equal to the number of integers in the list of integers minus one;

 index the list of integers in this pyramid by means of an indexing system; and

 providing an indication of indexing the list of integers in the hyperspace pyramid to thereby encode said list of integers to be encoded. Clause 2. The method claimed in clause 1, further comprising providing a list of alphanumeric codes and transforming the alphanumeric code list into a list of integers to be encoded.

Clause 3. The method claimed in clause 1, in which the indexing system indexes each of the plurality of vertices of said hyperspace pyramid. Clause 4. The method claimed in clause 1, in which the indication of the indexation of the list of integers in the hyperspace pyramid includes an integer representing the index of the projective point in the indexing system, m the degree total of said list of integers and n the number of integers in the list of integers.

Clause 5. The method claimed in clause 4, further comprising transforming the integer representing the index of the projective point by mx and nx combinatorial coordinates.

Clause 6. The method claimed in clause 4, further comprising the transformation of m the total degree of the list of integers and n the number of integers in the list of integers by an integer Y representing a global index.

Clause 7. The method claimed in clause 6, further comprising summing the integer Y representing a global index to the integer representing the index of the projective point to provide a total index Z, the indication of the indexation in the hyperspace pyramid including the total index Z. Clause 8. The method claimed in one of clauses 1 to 7, in which the list of integers is obtained from a file.

Clause 9. The method claimed in one of clauses 1 to 8, wherein the step of providing indexing indication in the hyperspace pyramid includes storing said indexing indication in a file.

Clause 10. The method claimed in one of clauses 1 to 8, wherein the step of providing indication of the indexation of the integer list in the hyperspace pyramid includes transmitting the indexing of the list of to a processing unit via a network. Clause 11. The method claimed in one of clauses 1 to 10 for encoding a list of integers, the method comprising providing the list of integers and generating the list of integers by means of the list of integers.

Clause 12. The method claimed in one of clauses 1 to 8, wherein the step of providing indication of the indexing of the integer list in the hyperspace pyramid includes displaying said indication of the indexation of the list of integers on a screen.

Clause 13. The use of the method claimed in clauses 1 to 3 and 4 to 7 to compress a string in which the string is associated with said list of integers.

Clause 14. A method for encoding a list of lists, the method comprising the iterative use of the method claimed in one of clauses 1 to 3 and 4 to 7 for encoding a list of lists.

Clause 15. The use of the method claimed in clause 1 to encrypt a list of integers, in which the indexing of the list of integers corresponds to the encryption of the list of integers. Clause 16. A method for decoding information encoded by the method for encoding information claimed in any of clauses 1 to 7, the method comprising:

 obtain the indication of the indexing of the list of integers in the hyperspace pyramid;

 obtaining an indication of an indexing system used to generate said indication of the indexing of the integer list in the hyperspace pyramid;

 determining by means of indexing and indicating an indexing system used said list of integers.

 provide said list of integers.

Clause 17. A computer comprising:

 a display unit;

 a processor;

 a data storage unit comprising a program configured to be executed by the processor, the program for encoding information, the program comprising:

 instructions for obtaining a list of integers to code;

 instructions for determining a hyperspatial pyramid having a size suitable for encoding the integer list, the hyperspace pyramid having a plurality of vertices the number of which is determined by the degree of the hyperspace pyramid which is equal to the sum of the integers in the list of integers and by the dimension of the hyperspatial pyramid which is equal to the number of integers in the list of integers minus one;

 instructions for indexing the list of integers in this pyramid by means of an indexing system;

 instructions for providing an indication of indexing the list of integers in the hyperspace pyramid to thereby encode said list of integers to be encoded.

Clause 18. An information storage device for storing executable instructions by a processor, which when executed will cause the processor to execute a method of encoding information, the method comprising obtaining a list of integers to code; determine a hyperspatial pyramid with a suitable size to encode the list of integers, the hyperspace pyramid having a plurality of vertices whose number is determined by the degree of the hyperspace pyramid which is equal to the sum of the integers of the integer list and the dimension of the hyperspace pyramid which is equal to the number of integers in the list of integers minus one; index the list of integers in this pyramid by means of an indexing system; and providing an indication of indexing the list of integers in the hyperspace pyramid to thereby encode said list of integers to be encoded.

Claims

claims

1. A method for encoding information, the method comprising:

 obtain a list of integers to code;

 determining a hyperspatial pyramid having a size suitable for encoding the list of integers, the hyperspace pyramid having a plurality of vertices the number of which is determined by the degree of the hyperspace pyramid which is equal to the sum of the integers of the integer list and by the dimension of the hyperspatial pyramid which is equal to the number of integers in the list of integers minus one;

 index the list of integers in this pyramid by means of an indexing system; and

 providing an indication of indexing the list of integers in the hyperspace pyramid to thereby encode said list of integers to be encoded.

The method claimed in claim 1, further comprising providing a list of alphanumeric codes and transforming the alphanumeric code list into a list of integers to be encoded.

The method claimed in claim 1, wherein the indexing system indexes each of the plurality of vertices of said hyperspace pyramid.

4. The method claimed in claim 1, wherein the indication of the indexing of the integer list in the hyperspace pyramid comprises an integer representing the index of the projective point in the indexing system, m the total degree of said list of integers and n the number of integers in the list of integers.

5. The method claimed in claim 4, further comprising transforming the integer representing the index of the projective point by mx and nx combinatorial coordinates.

The method claimed in claim 4, further comprising the transformation of m the total degree of the integer list and n the number of integers of the integer list by an integer Y representing a global index.

The method claimed in claim 6, further comprising summing the integer Y representing a global index to the integer representing the index of the projective point to provide a total index Z, the indication of the indexing. in the hyperspace pyramid comprising the total index Z.

8. The method claimed in one of claims 1 to 7, wherein the list of integers is obtained from a file.

The method claimed in one of claims 1 to 8, wherein the step of providing indexing indication in the hyperspace pyramid includes storing said indexing indication in a file.

The method claimed in one of claims 1 to 8, wherein the step of providing indexing indication of the integer list in the hyperspace pyramid includes transmitting the indexing of the integer list. to a processing unit via a network.

The method claimed in one of claims 1 to 10 for encoding a list of integers, the method comprising providing the list of integers and generating the list of integers using the list of integers.

The method claimed in one of claims 1 to 8, wherein the step of providing indication of the indexation of the integer list in the hyperspace pyramid includes displaying said indication of the indexing of the list of integers on a screen.

13. The use of the method claimed in one of claims 1 to 3 and 4 to 7 for compressing a string in which the character string is associated with said list of integers.

14. A method for encoding a list of lists, the method comprising iteratively using the method claimed in one of claims 1 to 3 and 4 to 7 for encoding a list of lists.

15. The use of the method claimed in claim 1 for encrypting a list of integers, wherein the indexing of the list of integers corresponds to the encryption of the list of integers.

16. A method for decoding information encoded by the method for encoding information claimed in one of claims 1 to 7, the method comprising:

 obtain the indication of the indexing of the list of integers in the hyperspace pyramid;

 obtaining an indication of an indexing system used to generate said indication of the indexing of the integer list in the hyperspace pyramid;

 determining by means of indexing and indicating an indexing system used said list of integers.

 provide said list of integers.

17. A computer comprising:

 a display unit;

 a processor;

 a data storage unit comprising a program configured to be executed by the processor, the program for encoding information, the program comprising:

 instructions for obtaining a list of integers to code;

instructions for determining a hyperspatial pyramid having a size suitable for encoding the integer list, the hyperspace pyramid having a plurality of vertices the number of which is determined by the degree of the hyperspace pyramid which is equal to the sum of the integers in the list of integers and by the dimension of the hyperspatial pyramid which is equal to the number of integers in the list of integers minus one; instructions for indexing the list of integers in this pyramid by means of an indexing system;

 instructions for providing an indication of indexing the list of integers in the hyperspace pyramid to thereby encode said list of integers to be encoded.

18. An information storage device for storing executable instructions by a processor, which when executed will cause the processor to execute a method for encoding information, the method comprising obtaining a list of integers to code; determining a hyperspatial pyramid having a size suitable for encoding the list of integers, the hyperspace pyramid having a plurality of vertices the number of which is determined by the degree of the hyperspace pyramid which is equal to the sum of the integers of the integer list and by the dimension of the hyperspatial pyramid which is equal to the number of integers in the list of integers minus one; index the list of integers in this pyramid by means of an indexing system; and providing an indication of indexing the list of integers in the hyperspace pyramid to thereby encode said list of integers to be encoded.