FR3060244A1 - METHOD AND DEVICE FOR CALCULATING A CORRECTING CODE - Google Patents

Abstract

A method of encoding a data vector into a transformed encoded vector, according to a linear error correcting code defined by a generator matrix (G), wherein: - the data vector and the generator matrix (G) are in the finite field of polynomials defined by: Bi = GF2 (x) / (pi (x)), where pi (x) is an irreducible polynomial that divides the polynomial xn + 1, i and n being integers, - The transformed encoded vector belonging to a ring and image by an isomorphism of the product of the generating matrix (G) by the data vector, the isomorphism, of the finite field Bi in a set Ai, being defined by: Fi (p (x)) = p (x) Qi (x), where A, is the principal ideal of the ring generated by the polynomial xn + 1 / pi (x), Qi (x) is the unique idempotent of said principal ideal, and p (x) ) a polynomial belonging to the body Bi, the ring being defined by: R2, n = GF2 (x) / (xn + 1), said method comprising: a step of calculating the image by the isomorphism Fi of the vector of given for to obtain a transformed data vector, - a step of determining a transformed matrix, in which each element located at a row and at a column is the sum of the image of the element located at said line and at said column. a raw matrix (G) by the isomorphism Fi and an element of the ring which has no component in the principal ideal, the raw matrix (G) being the generating matrix (G), - a step multiplying the matrix transformed by the transformed data vector to obtain said transformed encoded vector.

Description

Holder (s): HIGHER INSTITUTE OF AERONAUTICS AND SPACE Public establishment.

Extension request (s)

Agent (s): GEVERS & ORES.

PA) METHOD AND DEVICE FOR CALCULATING A CORRECTING CODE.

FR 3 060 244 - A1 (5 /) Method for encoding a data vector into a transformed encoded vector, according to a linear error correcting code defined by a generator matrix (G), in which:

- the data vector and the generating matrix (G) are in the finite body of polynomials defined by: Bi = GF ₂ (x) / (Pi (x)), Pi (x) being an irreducible polynomial which divides the polynomial x ^h +1, i and n being integers,

- The transformed encoded vector belonging to a ring and image by an isomorphism of the product of the generating matrix (G) by the data vector, the isomorphism, of the finite field B, in a set A ,, being defined by: F, (p (x)) = p (x) Qi (x), where A, is the main ideal of the ring generated by the polynomial x ⁿ +1 / ρ, (χ), Q / x) is l ' unique idempotent of said main ideal, and p (x) a polynomial belonging to the body B ,, the ring being defined by: R _{2 n} = GF ₂ (x) / (x ⁿ +1) said process comprising:

a step of calculating the image by the isomorphism F, of the data vector to obtain a transformed data vector,

- a step of determining a transformed matrix, in which each element located at a line and at a column is the sum of the image of the element located at said line and at said column of a raw matrix (G) by the isomorphism Fj and of an element of the ring which has no component in the main ideal, the raw matrix (G) being the generating matrix (G),

a step of multiplying the transformed matrix by the transformed data vector to obtain said transformed encoded vector.

i

METHOD AND DEVICE FOR CALCULATING A CORRECTING CODE

1. Technical field of the invention

The invention relates to correcting codes which allow the correction of errors in data, for example following their storage or their transmission. It relates in particular to the linear correcting codes defined by a generator matrix G of elements of a finite body. In such a correction code, the encoding of a data vector d of z elements of a finite body into an encoded vector c of t elements of this finite body, t> z, is obtained by multiplying the generating matrix G (of t rows and z columns) by the data vector d (c = G xd, or then c = dx G).

In particular, the correcting code can be a systematic code, that is to say, in which the first z elements of the encoded vector c are identical to the z elements of the data vector d. In this case, the notion of generating matrix can omit the first z lines which constitute an identity matrix, and the notion of encoded vector can omit z first elements of the encoded vector identical to the z elements of the data vector d.

In the event of an error during the storage or the transmission of a vector encoded c, there is then a vector to be corrected cf, identical to the vector encoded c except for the elements of c which are known to be erroneous, located at a index called "erroneous" from cf and which, in the following, are deleted from cf (typically a list of the erroneous indices is stored in addition to the elements of the vector). To find the data vector d from the vector to be corrected cf, we multiply the inverse of the generator matrix (in the case of a systematic code, we must reverse the generator matrix defined as comprising the first z lines which constitute a identity matrix) from which the lines of numbers identical to the erroneous indices of cf. have been deleted

2. Disadvantages of the state of the art

In the case where the volume of the data to be encoded or decoded is large, the abovementioned multiplications between a vector and a matrix necessary for encoding or decoding may require prohibitive computation times or resources.

3. Definitions

In the following, conventional objects and notations are used for those skilled in the art. However, certain definitions are recalled below.

Let GF ₂ (x) be the ring of polynomials with binary coefficients. Let ((x ⁿ + l)) be the ring generated by the polynomial x ⁿ + l, where n is an integer. Let (p, (x)) be the ring generated by an irreducible polynomial p, (x) of degree w which divides the polynomial x ⁿ + l, where i integer (ie: (Pi (x)) is the set of multiples of ρ, (χ)).

We denote B, = GF ₂ (x) / (Pî (x)), the body formed by the quotient between the ring GF ₂ (x) and the ring generated by the irreducible polynomial p, (x) of degree w .

We denote R _2n = GF ₂ (x) / (x ⁿ + l), the ring formed by the quotient between the ring GF ₂ (x) and the ring generated by the polynomial x ⁿ + l.

We also define the set A, as the main ideal of the ring R _2n generated by the polynomial χ ^η + 1 / ρ, (χ) (ie: a main ideal of a ring is an additive subgroup generated by a single element and which has the property of absorbing multiplication). We know that each element of the ring R2n can be expressed in the form of a sum of components (in other words of elements of the ring) where each of the components is in a principal ideal different from the ring R2n. We know that for any ideal A ,, there are 2 ^nw elements, including the zero element of the ring R2 n (ie: noted 0, in other words the additive neutral element) which have no component in l 'ideal A ,.

We recall that an idempotent is an element e (x) (here a polynomial) such that e (x) * e (x) = e (x). A, has a unique idempotent.

Classically, we say by extension that the vector i = (ii, i ₂ , ... i _r , ... ik) is the image by an isomorphism I of a vector ο = (θ!, Ο ₂ , ... o _r , ... o _k ), when i _r = l (o _r ), regardless of i belonging to {1,2, ..., k) (in other words when each element of the vector i of a given index is the image by the isomorphism I of the element of the same index in the vector o). We say that we compute the image by the isomorphism I of the vector o to obtain the vector i, when we perform, for all r belonging to (1,2, ..., k), the operation i _r = I (o _r ). The result of this calculation is the vector i (in other words each element of the vector i of a given index is obtained by the calculation of (and is equal to) the image by the isomorphism I of the element of the same index in the vector o).

Similar definitions are used for matrices ("index" is replaced by "row and column").

Classically, we will say that a vector or a matrix is in or belongs to a set if all the elements of this vector or of the matrix are in this set. The use of a so-called XOR-based representation (XOR means: or exclusive) is introduced in the following article: J. BLOEMER, M. KALFANE, M. KARPINSKI, R. KARP, M. LUBY AND D. ZUCKERMAN , D. An XOR-Based Erasure-Resilient Coding Scheme in Technical Report ICSI TR-95-048 (August 1995). It consists in representing each element by a binary square matrix where each binary number of this square matrix represents operations of XOR on binary numbers to be carried out for a multiplication with this element (for example in the multiplication of an element of the matrix with an element of a vector).

A polynomial is called AOP (or all-in-one polynomial, or in English "Ail One Polynomial") when the coefficients of all its monomials are equal to 1.

A polynomial g (x) which is written g (x) = x ^sr + x ^{s (r_1)} + x ^{s (r_2)} +. . . + x ^s + 1 = p (x ^s ), where p (x) is an AOP polynomial of degree r, is said to be ESP (or uniformly spaced polynomial, or in English “Equally Spaced Polynomial”, s and r are integers of course ).

4. Statement of the invention

To remedy the aforementioned drawbacks, the invention proposes a method of encoding a data vector, into a transformed encoded vector, according to a linear error correcting code defined by a generator matrix, in which:

- the data vector and the generating matrix are in the finite body of polynomials, denoted B ,, defined as follows: B, = GF ₂ (x) / (Pî (x)), p, (x) being a irreducible polynomial which divides the polynomial x ⁿ + l, i and n being integers,

- The encoded transformed vector belonging to the ring R _2n , and is the image by an isomorphism, noted F ,, of the product of the generating matrix by the data vector, the isomorphism F ,, of the finite field B, in the set A ,, the isomorphism being defined in the following way: F, (p (x)) = p (x) Qi (x) where A, is the main ideal of the ring R _2n generated by the polynomial x ⁿ + l / pj (x), Q / x) is the unique idempotent of said principal ideal A ,, and p (x) a polynomial belonging to the body B ,, the ring being defined as follows: R _2n = GF ₂ (x) / (x ⁿ + l) said method comprising the following steps:

- A step of calculating the image by the isomorphism F, of the data vector to obtain a transformed data vector,

- a step of determining a transformed matrix, in which each element of the transformed matrix (or more generally of a part of the transformed matrix comprising a certain number of rows of the transformed matrix) located at a line and at a column is the sum of the image of the element located at said line and at said column of a raw matrix by the isomorphism F, and of an element of the ring R _{2 n} which has no component in l 'principal ideal A ,, the raw matrix being the generating matrix,

a step of multiplying the matrix transformed by a transformed vector, the transformed vector being the vector of transformed data, to obtain said encoded vector transformed (thus therefore the encoded vector transformed is the result of the multiplication of the generator matrix transformed by the transformed data vector).

In one embodiment of the encoding method, the method comprises a step of calculating the image by an inverse isomorphism, denoted F ¹ , of the encoded vector transformed to obtain an encoded vector, the inverse isomorphism F, ¹ being l 'inverse of the isomorphism F, and is defined as follows: p (x) = pn (x) mod p, (x), pn (x) being a polynomial of the ring R _2n , the encoded vector being equal to the product of the generator matrix by the data vector. Thus, once the multiplication step has been carried out, the result can be passed back into the body B ,. As a variant, the result can remain in the ring R _2n so as to subsequently carry out the decoding without having to transform the vector to be corrected encoded towards the ring R _2n . Still in order to remedy the aforementioned drawbacks, the invention also relates to a method for decoding a vector to be corrected transformed into a data vector according to a linear error correcting code defined by a generator matrix, said vector to be corrected comprising a plurality of erroneous indicia comprising an erroneous element (in other words, comprising a plurality of erroneous elements situated at erroneous indices) in which:

- the data vector and the generating matrix are in the finite field B ,, defined as follows: B, = GF ₂ (x) / (pj (x)), ρ, (χ) being an irreducible polynomial which divides the polynomial x ⁿ + l, i and n being integers,

- The transformed vector to be corrected belonging to the ring R _2n , defined as follows: R _2n = GF ₂ (x) / (x ⁿ + l) said process comprising the following steps:

a step of determining a transformed matrix, each element of the transformed matrix (or more generally of a part of the transformed matrix comprising a certain number of rows of the transformed matrix) located at a line and at a column is the sum of the image of the element located at said line and at said column of a raw matrix by the isomorphism F, and of an element of the ring R _2n which has no component in the main ideal A ,, the raw matrix being the inverse of the generating matrix from which the lines corresponding to said plurality of erroneous indices are deleted, the isomorphism F ,, of the finite body B, in a set A ,, being defined from the as follows: F, (p (x)) = p (x) Qj (x), where A, is the main ideal generated by the polynomial x ⁿ + l / pj (x) of the ring R _2n , Ο ,, (χ) is the unique idempotent of said principal ideal A ,, and p (x) a polynomial belonging to the body B ,,

a step of multiplying the transformed matrix by a transformed vector, the transformed vector being the vector to be corrected transformed, to obtain a transformed data vector.

a step of calculating the image by an inverse isomorphism, denoted F, ¹ , of the data vector transformed to obtain the data vector, the inverse isomorphism F, ¹ being the inverse of the isomorphism F, and is defined as follows:

p (x) = pn (x) mod p, (x), pn (x) being a polynomial of the ring R _2n .

In one embodiment of the decoding method, the method comprises, prior to the multiplication step, a step of calculating the image by the isomorphism F, of a vector to be corrected to obtain the transformed vector to be corrected, the vector of data being equal to the product of the inverse of the generating matrix from which the lines of the same number as said plurality of erroneous indices are deleted, by the vector to be corrected. In this case, the decoding method is compatible with an encoding method where the result of the multiplication step is passed back into the body B i.

Below are advantages and optional features common to both the encoding process and the decoding process.

By thus carrying out the multiplication step in the ring R _2n where the multiplications between elements can be faster than in the body B ,, the invention makes it possible to accelerate the multiplication step.

According to the invention (decoding or encoding), the element of the ring R _2n which has no component in the main ideal A, can either be zero or be non-zero. In the case where it is zero, the element of the transformed matrix is the image of the element located at the same line and at the same column of a raw matrix (G) by the isomorphism Fj.

When all the elements of the transformed matrix are in this case, the transformed matrix is the image of the raw matrix by the isomorphism F,.

As a variant, at least one of the elements of the transformed matrix located at a line and at a column is the sum of the image of the element located at said line and at said column of the raw matrix by the isomorphism F, and of a non-zero element of the ring R _2n which has no component in the main ideal A ,. Thus, the element of the ring R _2n (zero or not), which has no component in the main ideal A ,, can be chosen so as to minimize the number of XOR generated by the element of the transformed matrix thus formed in a multiplication with another element (in a so-called XOR-based representation, it is the quantity of binary numbers equal to 1 in the representation of this element of the transformed matrix).

This thus makes it possible to reduce the time or the resources necessary for the multiplication stage.

In one embodiment, the encoding method or the decoding method is implemented in a so-called or exclusive representation in which each element of the transformed matrix is represented by an elementary square matrix of n rows and n columns of binary numbers.

According to a particular implementation of this embodiment, said each element is stored in the form of a single one of these n rows or n columns, the transformed matrix thus being stored in the form of a compressed transformed matrix consisting of the single n rows or n columns of each element of the transformed matrix.

Thanks to the passage in the ring R _2n , in a representation based on XOR, the elements of the transformed matrix have only diagonals made up either of 1, or only of 0. This is why, we can allow ourselves to memorize only one row or one column for each element. This saves memory space.

In one embodiment (for both the decoding method and the encoding method), each element of the transformed vector is represented by an elementary vector of n binary numbers, said step of multiplying the matrix transformed by the transformed vector comprising a substep of multiplication of the elementary square matrix by the elementary vector, said substep of multiplication comprising the following steps:

a step of reading from memory said single row or single column of the elementary square matrix in the compressed transformed matrix,

- A processing step in which, for each binary number occupying a position of said one single row or single column in the compressed transformed matrix, if said bit is equal to a determined value (the determined value is often equal to 1. So general, it is about a value which defines an XOR which can of course be different from 1 according to the implementations), or exclusive generated by all the diagonal of the elementary square matrix determined by this position are calculated.

This reduces the computation time necessary for carrying out the encoding process or the decoding process since it suffices to read only one row or column per element to carry out the multiplication sub-step.

In one embodiment, the encoding method or the decoding method can comprise, prior to the multiplication step, the following steps:

- a step of determining an identical sub-part comprising we binary elements (having a value defining an XOR. Classically this value is 1) in m lines of the compressed transformed matrix, we greater than or equal to two,

a step of calculating at least one intermediate result from said identical sub-part,

- the multiplication step being carried out from the intermediate result so as to avoid n. (we.m- (we + m)) or exclusive operations during the multiplication step.

By thus identifying an identical sub-part, common XORs involved in a plurality of multiplication sub-steps are determined so as to calculate these XOR only once to carry out the plurality of multiplication sub-steps.

The sub-parts can be aligned or offset cyclically in the transformed matrix. The use of the compressed transformed matrix makes it possible in particular to find offset sub-parts, the search for offset sub-parts being too complex and too long in the case of the use of a transformed matrix without compression.

The polynomial ρ, (χ) can be an AOP polynomial or an ESP polynomial.

In this case, two approaches allow a fast computation of the images by the isomorphism F, and the inverse isomorphism F, ¹ .

According to a first approach:

- each element of the finite field B, being represented by a vector of binary data

- the computation of the image of each element of the finite body B, by the isomorphism F, includes the addition of a bit (or binary data) of parity to said vector of binary data.

- the calculation of the image of each element of the finite body B, by the inverse isomorphism F, ¹ (inverse of the isomorphism F,) includes the removal of said parity bit to said vector of binary data.

According to a second approach:

- each element of the finite field B, being represented by a vector of binary data

- the computation of the image of each element of the finite body B, by the isomorphism F, includes the addition of a binary datum (or bit) equal to the null element (in other words, the additive neutral element) at a position of said binary data vector.

- the calculation of the image of each element of the finite body B, by the inverse isomorphism F, ¹ (inverse of the isomorphism F,) includes the addition of the binary data located at said position to a plurality of binary data said vector of binary data, then deleting said binary data located at said position of said vector of binary data.

The invention also relates to a device suitable for implementing the steps of the encoding or decoding method.

The invention also relates to:

A computer program comprising instructions adapted to the implementation of each of the steps of the encoding method or of the decoding method when said program is executed on a computer, and

An information storage means, removable or not, partially or totally readable by a computer or a microprocessor comprising code instructions of a computer program for the execution of each of the steps of the encoding method or of the method decoding.

Other advantages and optional features of the encoding method are presented below.

According to one embodiment, the step of determining the transformed matrix comprises the following steps:

- a step to generate a transformed candidate matrix,

a step for estimating a number of or exclusive operations to be carried out for the step of multiplying the transformed matrix by the transformed vector, when the transformed candidate matrix is chosen as a transformed matrix,

a step for determining, as a function of said number of operations of or exclusive to be carried out, whether the transformed candidate matrix is chosen as the transformed matrix.

In this way, we obtain the transformed matrix which allows the least costly multiplication step possible in computation time.

According to a particular implementation of this embodiment, the transformed candidate matrix can consist of elements in the ring R ₂ , _n which are, in a so-called XOR-based representation, cyclic shifts of the identity matrix ( of n rows and n columns).

In particular, the encoding method can include:

- during the step to estimate the number of operations of or exclusive to be carried out for the step of multiplying the matrix transformed by the transformed vector, a step of determining an identical sub-part comprising w binary elements in m lines of the compressed transformed matrix, w greater than or equal to two,

- a step of calculating the number of operations of or exclusive avoided, equal to n. (we.m (we + m)), from said sub-parts, in which the number of operations of or exclusive to be carried out is estimated from said number of exclusive or exclusive operations avoided.

One thus takes into account, in the number of estimated operations, of the simplification of computations carried out thanks to the search for identical under parts.

In a first embodiment, the transformed candidate matrix is obtained by the following steps:

- a step to generate a candidate generator matrix defining an error correcting code which is MDS (Maximum Separable Distance).

- a step of calculating the image by the isomorphism Fj of the generating candidate matrix to obtain the transformed candidate matrix

As a variant, the transformed candidate matrix is obtained by the following steps:

a step of calculating the image of the candidate generator matrix by the inverse isomorphism F, ¹ to obtain the candidate generator matrix,

-A verification step that the candidate generator matrix is MDS.

5. List of figures

Other objects, characteristics and advantages of the invention will appear on reading the following description given by way of non-limiting example and which refers to the appended figures in which:

FIG. 1 represents a device suitable for implementing the method of FIG. 2.

FIG. 2 represents the steps of a method according to an embodiment of the invention.

Figure 3 shows the detail of a step of Figure 2, in one embodiment.

FIG. 4 represents a step of FIG. 3 in one embodiment and FIG. 5 represents this same step in a second embodiment.

Figure 6 shows in detail a step in Figure 2 and the storage of the transformed matrix.

Figures 7, 8, 9 and 10 give examples for the isomorphism Fj and its inverse F, ¹ .

6. Detailed description of an embodiment of the invention

The following embodiments are examples. Although the description refers to one or more embodiments, this does not necessarily mean that each reference relates to the same embodiment, or that the characteristics apply only to a single embodiment. Simple features of different embodiments can also be combined to provide other embodiments. In the figures, the scales and the proportions are not strictly observed, for the purposes of illustration and clarity.

The device of FIG. 1 comprises a memory 200, hard disks and 300 to 306 in which a microprocessor 100 can read and write data via a data bus 400. The memory 200 (for example a removable disk or not, or else a memory in the form of an integrated circuit (for example of the flash EEPROM type)) comprises, in a part 250, the code instructions of a computer program for the execution of each of the steps of the method which will be described with reference to FIG. 2. An input / output 500 is also connected to the microprocessor 100 via the bus 400. Other architectures are of course possible.

FIG. 2 describes the steps carried out by the microprocessor 100 in an embodiment of the invention. The microprocessor includes a working memory (not shown) used in the process steps below.

In step 1000, the microprocessor 100 obtains the data vector d for example by receiving it by the input / output 500 and via the bus 400. The data vector d consists of z elements (z being obviously an integer ) of body B, defined above.

In step 2000, the microprocessor 100 calculates the image of the vector d by the isomorphism F, defined above and stores the result in the vector dt consisting of z elements in the ring R _2n .

In step 3000, a transformed matrix GT is obtained by the microprocessor 100. Each element of this transformed matrix GT located at a line and at a column is the sum of the image of the element located at said line and at said column of a generator matrix G by the isomorphism F, and of an element of the ring R _2n which has no component in the main ideal A, defined above.

In general, in one embodiment, in step 3000, the transformed matrix GT, can be previously determined and stored in a memory, which may be non-volatile, for example, in the embodiment of FIG. 2 , in memory 200 or in own memory of processor 100. The transformed matrix

GT is then obtained, for example by the processor 100, by reading this memory, for the encoding of a plurality of vectors. FIG. 3 illustrates an embodiment for predetermining the transformed matrix. As a variant, the transformed matrix GT can be recalculated each time that the transformed generator matrix is obtained, for example by applying the isomorphism F, defined above to the generator matrix G itself stored.

The generator matrix G is in the body B ,, and defines a linear correction code. It can be made up of t rows and z columns (t greater than z, the first z rows of G can conventionally constitute an identity matrix). The transformed generator matrix GT, in the ring R _2n , naturally has the same number of rows and columns as the generator matrix G.

In step 4000, the transformed matrix GT is multiplied by the data vector dt. The result of this multiplication is the encoded vector transformed and made up of t elements in the ring R _2n .

In step 5000, the reverse isomorphism denoted F, ¹ defined above is applied to the encoded vector transformed and. The result is a vector encoded c. the encoded vector c is equal to the product of the generating matrix G by the data vector d, in other words the result of the encoding of the data vector according to the linear correcting code defined by the generating matrix G. The encoded vector c understands well heard t elements of body B ,.

In step 6000, the encoded vector c is divided into seven data blocks (cbO, cb2, cb3, cb4, cb5, cb6) by the microprocessor 100 which stores each of these seven blocks in each of the seven disks 300, 301, 302, 304, 305, 306. Here the number seven for the number of blocks and the number of disks is given of course by way of example, and perhaps any. At step 6000 ′, the disk 302, for example, is out of service following an incident which takes place after these blocks have been stored in step 6000. By way of example, this incident has the consequence that the cb2 data block is considered to be erased, lost or erroneous for the indices between r and s which will be said to be erroneous. The resulting vector is the vector to be corrected cf.

In step 7000, the image of the vector to be corrected cf is calculated by the isomorphism F, to obtain the vector to be corrected transformed cft.

At step 8000, one obtains a transform matrix G ^-1 T. Each element of the transform matrix G obtained ^_1 T, located in a line and a column is the sum of the image of the element at said line and to said column of a raw matrix by the isomorphism F, and of an element of the ring R _2n which has no component in the main ideal A ,, the raw matrix being the reverse (G ¹ ) from the generator matrix G from which the lines corresponding to said plurality of erroneous indices are deleted (in our example the (numbers of) lines between r and s).

In general, during decoding, (and not only in this embodiment), the transformed matrix G ^_1 T can be obtained either:

- by applying the isomorphism F, unlike G ¹ of the generator matrix G from which inverse are deleted the lines corresponding to the erroneous indices and possibly by adding, to certain elements of the matrix, an element of the ring R _2n which has no component in the main ideal A, (analogously to what is described below in step 3100),

- from the inverse of the transformed matrix GT used for encoding (in the embodiment of FIG. 2 determined in step 3000, and used to encode the data vector in step 4000), of which inverse G ¹ we deleted the lines of the same number as the erroneous indices.

In step 9000, a transformed data vector dt is obtained from the multiplication of the transformed matrix G ^_1 T by the transformed vector to be corrected cft.

In step 10000, the image of the transformed data vector dt is calculated by applying the inverse isomorphism F, ¹ defined above. The result is the data vector d. The erroneous block, (in other words erased or lost), in the embodiment of FIG. 2 cb2, was thus found through the use of the linear error correcting code defined by the generator matrix G.

The data vector d is then encoded again into encoded vector c and stored for example in a new disc 302.

Steps 5000 and 7000 can be omitted together. It is indeed possible to store in disks 300 to 306 in step 6000 the encoded vector transformed and (in data blocks), which is in the ring R _2n , instead of storing the encoded vector c.

The encoding and decoding methods are described with reference to FIG. 2 and are implemented by the same device in FIG. 1. However, in practice, the encoding method and the decoding method can of course be carried out. independently and by different machines.

The microprocessor 100 can be replaced by a computer, for example by an internet server, or else be included in an internet server as well as the memory 200, the disks 300 to 306 and the bus 400.

The invention does not only apply to the encoding and decoding of data in memory, for example on a hard disk. It can also be applied to the encoding and decoding of data for communication on a communication channel, for example radio.

In known manner, the encoded vector c is equal to the product of the generating matrix G (defining the linear correcting code) and the data vector d, and the data vector d is equal to the product of the inverse G ¹ of the matrix generator, from which the lines of the same number as the erroneous indices of the vector to be corrected have been deleted.

Figure 3 illustrates the steps to obtain the transformed generator matrix GT. In step 3500, a transformed candidate matrix GTc in the ring R _{2 n} is generated. Figures 4 and 5 will give further details of this step, each according to an embodiment of the invention.

At step 3600, the microprocessor 100 estimates a number of or exclusive operations to be carried out for the step of multiplying the transformed matrix GT by the transformed vector dt when the transformed candidate matrix GTc is chosen as the transformed matrix GT . In a representation called based on or exclusive, the number of operations of or exclusive is obtained from the total number of 1 in the elements of the ring R _2n of the transformed candidate matrix GTc (or of the transformed matrix GT) .

In step 3700, the processor 100 determines whether the transformed candidate matrix GTc is chosen as the transformed matrix. In general, in one embodiment of the invention, a plurality of transformed candidate matrices GTc are generated (for example from random numbers). The transformed candidate matrix GTc for which the number of operations is the smallest, among the plurality of matrices, is chosen.

FIG. 4 now illustrates how the transformed candidate matrix GTc can be generated in particular during step 3500, in a first embodiment. In one embodiment, the transformed matrix GT can also be obtained directly in this way in the step of determining the transformed matrix (in step 3000 in the embodiment of FIG. 2) without performing steps 3600 and 3700.

In step 3100, a candidate candidate generator matrix Gc in the body B ,, defining a correction code MDS (Maximum Distance Separable) is generated, for example (in an embodiment of the invention) using known methods to build a generalized Cauchy, Vandermonde or Cauchy matrix also known per se.

In step 3200, the microprocessor 100 uses the isomorphism F, to obtain the transformed candidate matrix GTc from the generator candidate matrix Gc. In a first embodiment, the image is computed by the isomorphism F, of the generator candidate matrix Gc to obtain the transformed candidate matrix GTc. As a variant, we add, for at least one element of the transformed candidate matrix (GTc), to the image of the elements of generator candidate matrix Gc, a non-zero element of the ring R _2n which has no component in the main ideal A ,, for example when this addition makes it possible to reduce the number of 1 (we therefore place ourselves here in a so-called XOR-based representation of course) in the element of the transformed candidate matrix. In other words, in an embodiment of the invention, an element of the transformed matrix (GT, G ^_1 T) (or in particular, for this figure, of the transformed candidate matrix) located at a line and at a column is the sum of the image of the element located at said line and at said column of the raw matrix (here the generator matrix or the candidate generator matrix) by the isomorphism F, and of an element of the ring R _2n nonzero which has no component in the main ideal A, for example, when this element located at said line and at said column of the transformed matrix has a number of 1 less than the number of 1 present in the image of l element located at said line and at said column of the raw matrix. In this embodiment, the invention then makes it possible to obtain sparse transformed matrices, which reduces the number of or exclusive operations to be carried out for the step of multiplying the matrix transformed by the transformed vector, when the transformed candidate matrix is chosen as transformed matrix.

The elements in the ring R _2n which have no component in the main ideal A ,, are the multiples of ρ, (χ) in this ring, and each element can be obtained by calculating at least part of these multiples.

FIG. 5 now illustrates how the transformed candidate matrix (s) GTc can be generated in particular during step 3500, in a second embodiment. In step 3300, a transformed candidate matrix GTc is generated. According to one embodiment of the invention, the transformed candidate matrix can consist of elements in the ring R _2n which are, in a so-called XOR-based representation, cyclic shifts of the identity matrix (of n lines and n columns). At step 3400, the image of the transformed candidate matrix GTc by the inverse isomorphism F, ¹ (previously defined) is calculated to obtain a generator candidate matrix Gc (we can choose to subtract from the image, for all or part of the elements of the transformed candidate matrix, a non-zero element of the ring R _{2 n} which has no component in the main ideal A,). If the generating candidate matrix is MDS, the transformed candidate matrix GTc is retained (for example to carry out steps 3600 and 3700 in FIG. 3). Otherwise, the transformed candidate matrix GTc is rejected.

FIG. 6 presents in more detail step 4000 (step 9000 can be carried out in the same way) and describes how the transformed matrix is memorized. FIG. 6 presents a part of a transformed matrix GT (or else the whole of a transformed matrix, for example when the notion of generating matrix excludes the first z lines which constitute an identity matrix in a systematic code) comprising three lines and three columns , in the case where n = 5, and where the transformed matrix is stored in a so-called or exclusive representation. Each element in the ring R ₂₅ of the transformed matrix GT is an elementary square matrix of 5x5 binary numbers. The smallest boxes represent these binary numbers. Those comprising a point or a cross are those which memorize 1. The others memorize 0. The figure represents in particular a part of the step 4000 of multiplication of a part of the transformed matrix GT by a part of the data vector transformed dt consisting of the vectors aO, al, a2 in the ring R ₂ , ₅ to obtain the corresponding part of the encoded vector transformed and consisting of the vectors pO, pl, p2 in the ring R _2j5 (in the case of a code systematic where we omit the z elements identical to the data vector, dt and and can represent respectively the whole of the data vector and the whole of the encoded vector transformed).

The transformed matrix GT is for example stored in the form of a compressed matrix GT 'in which each element of the transformed matrix GT is stored in the form of its first line for example 610. This is possible because the elements in the ring all have diagonals full of 1 or full of 0.

Instead of using the first row, you can use other rows or a column. The use of such a compressed matrix makes it possible to reduce the memory space and the computation time necessary for carrying out the method of FIG. 2 or more generally of the invention.

In particular, step 4000 comprises a substep of multiplication of the elementary square matrix 610 by the vector a _o of dt during the calculation of the vector p ₀ of and. This step can be carried out by reading the line 610 without reading the rest of the elementary square matrix 600 in memory. During this reading, if a binary number, for example 612 is worth 1 in this line 610, all the XOR generated by the diagonal determined by the position of 612 are carried out, for example the XOR determined by the fact that the binary number 613 is at 1.

In addition, to limit the number of XORs produced during step 4000, an identical sub-part is sought in the three lines of the part of the compressed transformed matrix GT ′. In the three lines of GT ', the sub-part between the two binary numbers represented by a box comprising a point is identical (we do not take into account the 1 present between these two binary numbers). In general, the identical sub-part defines identical XORs involved in the calculation of several elements of the vector and. In the example of FIG. 6, the three identical sub-parts define XORs which are involved in the calculation of the three elements p ₀ , Pi and p ₂ of the vector and. Generally speaking, an intermediate result is therefore calculated which represents the number of XORs determined by the identical sub-part and this intermediate result is used for the calculation of other elements of the transformed vector. Thus, if the identical sub-part includes we binary elements (ie: equal to 1, or more generally representing an XOR) and if the identical sub-part is present in m lines, we save n (we * m- (we + m) ) XOR. In the example given in Figure 6, we thus save 5 XOR.

Figure 6 gives an example where the occurrences of the identical part are aligned in G '(ie: in the identical part, according to the lines, the binary numbers at 1 are in the same columns), but these can be more generally, in one embodiment of the invention, offset (ie: in the identical part, along the lines, the binary numbers at 1 (or more generally representing an XOR in a multiplication with another element) are in different columns, the quantity of binary numbers (0 or 1) between each pair of 1 of the identical part being identical). The use of the compressed matrix makes it possible to search in a realistic time for such identical offset sub-parts, thus making it possible to further reduce the number of XORs produced during step 4000. According to one embodiment, the polynomial p, ( x) is an AOP polynomial or an ESP polynomial.

In a first variant of this embodiment, the application of the isomorphism F includes the addition of a parity bit to the element of the body, as illustrated in FIGS. 7 and 8. The application of the inverse isomorphism F, ¹ then consists in suppressing the parity bit.

Figure 7 represents the computation of the image by the isomorphism Fj (x) for n = 5 for a polynomial p, (x) AOP. According to this variant, for an AOP polynomial, the application of the isomorphism F, consists in adding a parity bit (i.e.: result of XOR) of all the binary numbers constituting the element of the body.

FIG. 8 represents the computation of the image by the isomorphism F, for a polynomial p, (x) ESP. It can be shown that the ideal corresponding to the finite field determined by an ESP is the set of elements which can be decomposed as an interlacing of s words of even weight of length r + 1. In this embodiment, the calculation of the image of an element consisting of sr bits forming r blocks of s consecutive bits (in FIG. 8, r = 2, s = 3, block 1 and block 2 form l element) consists in adding a block of s bits (in figure 2, block 3) which is the XOR (bit by bit) of the r blocks of s bits of the element (in figure 2 which is the XOR of block 1 and block 2. The image is formed by blocks 1, 2 and 3). In a second variant of this embodiment, the computation of the image of each element of the finite body B, by the isomorphism F, comprises the addition of a binary datum equal to 0 at a position of said bit vector.

As illustrated in FIGS. 9 and 10, the computation of the image of each element of the finite body B, by inverse isomorphism F, ¹ includes the addition of the binary data located at said position to a plurality of binary data of said vector binary data, then deleting said binary data located at said position of said bit vector.

Figure 9 represents the computation of the image by the inverse isomorphism F, ¹ for n = 5 for a polynomial p, (x) AOP. In this embodiment, in general, the computation of the image of each element of the finite body B, by the inverse isomorphism Fj ¹ consists in adding the binary datum located at said position to each binary datum of said vector binary data, then deleting said binary data located at said position of said bit vector. Figure 10 represents the computation of the image by the inverse isomorphism noted F ^ for a polynomial ESP. This calculation consists in making XOR operations between the last block of s bits and each of the r blocks of s bits of the element. In Figure 10, s = 3 and r = 2.

Claims (20)

1. Method for encoding a data vector (d), into a transformed encoded vector (and), according to a linear error correcting code defined by a generating matrix (G), in which: - the data vector (d) and the generating matrix (G) are in the finite field of polynomials, denoted B ,, defined as follows: B, = GF ₂ (x) / (Pî (x)), p , (x) being an irreducible polynomial which divides the polynomial x ⁿ + l, i and n being integers, - The encoded transformed vector (and) belonging to a ring, noted R _2n , and is the image by an isomorphism, noted F ,, of the product of the generating matrix (G) by the data vector (d), the isomorphism Fj, of the finite field B, in a set A ,, being defined as follows: F, (p (x)) = p (x) Ο ,, (χ), where A, is the principal ideal of the ring R _2n generated by the polynomial χ ^η + 1 / ρ, (χ), Ο ,, (χ) is the unique idempotent of the main ideal A ,, and p (x) a polynomial belonging to the body B ,, the ring being defined as follows: R _2n = GF ₂ (x) / (x ⁿ + l) said process comprising the following steps: - A step of calculating the image by the isomorphism F, of the data vector (d) to obtain a transformed data vector (dt), a step of determining a transformed matrix (GT), in which each element of the transformed matrix (GT) located at a line and at a column is the sum of the image of the element located at said line and at the said column of a raw matrix (G) by the isomorphism Fj and of an element of the ring R _{2 n} which has no component in the main ideal A ,, the raw matrix (G) being the matrix generator (G), a step of multiplying the transformed matrix (GT) by a transformed vector (dt), the transformed vector (dt) being the vector of transformed data (dt), to obtain said encoded vector transformed (and). 2. The encoding method according to claim 1 comprising a step of calculating the image by an inverse isomorphism, denoted Fj ¹ , of the transformed encoded vector (and) to obtain an encoded vector (c), the inverse isomorphism Fj ¹ being the inverse of the isomorphism F, and is defined as follows: p (x) = pn (x) mod p, (x), pn (x) being a polynomial of the ring R _2n , the vector encoded (c) being equal to the product of the generator matrix (G) by the data vector (d). 3. Method for decoding a vector to be corrected transformed (cft), into a data vector (d) according to a linear error correcting code defined by a generator matrix (G), said vector to be corrected comprising a plurality of erroneous clues comprising an erroneous element in which: - the data vector (d) and the generating matrix (G) are in the finite body of polynomials, noted B ,, defined as follows: B, = GF ₂ (x) / (pj (x)), ρ , (χ) being an irreducible polynomial which divides the polynomial x ⁿ + l, i and n being integers, - The transformed vector to correct (cft) belonging to a ring, denoted R _2n , defined as follows: R ₂ , _n = GF ₂ (x) / (x ⁿ + l) said process comprising the following steps: - a step of determining a transformed matrix (G ^_1 T), each element of the transformed matrix (G ^_1 T) located at a line and at a column is the sum of the image of the element located at said line and to said column of a raw matrix (G ¹ ) by the isomorphism F, and of an element of the ring R2n which has no component in the main ideal A ,, the raw matrix being the reverse (G ¹ ) of the generating matrix (G) from which the lines corresponding to said plurality of erroneous indices are deleted, the isomorphism F ,, of the finite field B, in a set A ,, being defined as follows: F, (p (x)) = p (x) Qj (x), where A, is I 'principal ideal generated by the polynomial x ⁿ + l / Pi (x) of the ring R2n, Ο ,, (χ ) is the unique idempotent of said principal ideal A ,, and p (x) a polynomial belonging to the body B ,, a step of multiplying the transformed matrix (G ^_1 T) by a transformed vector (cft), the transformed vector (cft) being the vector to be corrected transformed (cft), to obtain a transformed data vector (dt), a step of calculating the image by an inverse isomorphism, denoted F, ¹ , of the transformed data vector (dt) to obtain the data vector (d), the inverse isomorphism F, ¹ being the inverse of l isomorphism F, and is defined as follows: p (x) = pn (x) mod p, (x), pn (x) being a polynomial of the ring R _{2 n} . 4. decoding method according to claim 3, comprising, prior to the multiplication step, a step of calculating the image by the isomorphism F, of a vector to be corrected (cf) to obtain the vector to be corrected transformed (cft), the data vector (d) being equal to the product of the inverse (G ¹ ) of the generator matrix (G) from which are deleted the lines of the same number as said plurality of erroneous indices_to said plurality of erroneous indices by the vector to be corrected (cf). 5. Method according to one of claims 1 to 4 in which, the transformed matrix (GT, G ^_1 T) is the image of the raw matrix (G, G ¹ ) by the isomorphism F, 6. Method according to one of claims 1 to 4, wherein, at least one of the elements of the transformed matrix (GT, G ^_1 T) located at a line and at a column is the sum of the image of the element located at said line and at said column of the raw matrix (G, G ¹ ) by the isomorphism F, and of a non-zero element of the ring R _2n which has no component in the main ideal A , 7. Method according to any one of the preceding claims, implemented in a so-called or exclusive representation in which each element of the transformed matrix (GT, G ^_1 T) is represented by an elementary square matrix of n lines and n binary number columns, said each element being stored in the form of only one of these n rows or n columns, the transformed matrix thus being stored in the form of a compressed transformed matrix consisting of only one of the n rows or n columns of each element of the transformed matrix (GT, G ^_1 T). 8. Method according to the preceding claim in which each element of the transformed vector (dt, cft) is represented by an elementary vector of n binary numbers, said step of multiplying the transformed matrix (GT, G ^_1 T) by the transformed vector ( dt, cft) comprising a sub-step of multiplication of the elementary square matrix by the elementary vector, said sub-step of multiplication comprising the following steps: a step of reading from memory said single row or single column of the elementary square matrix in the compressed transformed matrix, a processing step in which, for each binary number occupying a position of said one single row or single column in the compressed transformed matrix, if said bit is equal to a determined value, the or exclusives generated by the entire diagonal of the matrix elementary square determined by this position are calculated. 9. Method according to claim 8, comprising, prior to the multiplication step, the following steps: a step of determining an identical sub-part comprising we binary elements in m lines of the compressed transformed matrix, we greater than or equal to two, - a step of calculating an intermediate result from said identical sub-part, - the multiplication step being carried out from the intermediate result so as to avoid n. (we.m- (we + m)) operations of or exclusive during the multiplication step. 10. Method according to any one of the preceding claims dependent on claim 1 comprising the following steps: - a step to generate a transformed candidate matrix, - a step to estimate a number of operations of or exclusive to perform for the step of multiplying the transformed matrix (GT) by the transformed vector (dt, cft), when the candidate candidate matrix is chosen as the transformed matrix (GT), a step for determining, as a function of said number of operations of or exclusive to be carried out, whether the transformed candidate matrix is chosen as a transformed matrix (GT). 11. The method of claim 10 taken in dependence on claim 7 further comprising: - during the step to estimate the number of operations of or exclusive to be carried out for the step of multiplying the transformed matrix (GT) by the transformed vector (dt, cft), a step of determining an identical sub-part comprising w binary elements in m lines of the compressed transformed matrix, w greater than or equal to two, - a step of calculating the number of operations of or exclusive avoided, equal to n. (we.m (we + m)), from said sub-parts, in which the number of operations of or exclusive to be carried out is estimated from said number of exclusive or exclusive operations avoided. 12. Method according to any one of claims 10 and 11 further comprising: a step to generate a candidate candidate matrix defining an error correcting code which is MDS. - a step of calculating the image by the isomorphism Fj of the generating candidate matrix to obtain the transformed candidate matrix 13. Method according to any one of claims 10 and 11 comprising the following steps: a step of calculating the image of the candidate generating matrix by an inverse isomorphism F, ¹ to obtain the candidate generating matrix, the inverse isomorphism F, ¹ being the inverse of the isomorphism F, and is defined from the as follows: p (x) = pn (x) mod ρ, (χ), pn (x) being a polynomial of the ring R _2n . -a verification step that the candidate generator matrix is MDS. 14. Method according to any one of the preceding claims, in which the polynomial ρ, (χ) is an AOP polynomial. 15. Method according to any one of the preceding claims, in which the polynomial p, (x) is an polynomial ESP. 16. Method according to any one of claims 14 or 15, in which - each element of the finite field B, being represented by a vector of binary data - the computation of the image of each element of the finite body B, by the isomorphism F, includes the addition of a bit of parity to said vector of binary data. the computation of the image of each element of the finite body B, by an isomorphism F, ¹ includes the removal of said parity bit from said binary data vector, 5 the inverse isomorphism F, ¹ being the inverse of the isomorphism F, and is defined as follows: p (x) = pn (x) mod p, (x), pn (x) being a polynomial of the R _2n ring. 17. Method according to any one of claims 14 or 15, in which - each element of the finite field B, being represented by a vector of binary data 10 - the computation of the image of each element of the finite body B, by the isomorphism F, comprises the addition of a binary datum equal to 0 at a position of said bit vector. the computation of the image of each element of the finite body B, by an inverse isomorphism denoted Fj ¹ comprises the addition of the binary data located at said position to a plurality of binary data of said vector of binary data, then the deletion of said 15 binary data located at said position of said bit vector, the inverse isomorphism F, ¹ being the inverse of the isomorphism F, and is defined as follows: p (x) = pn (x) mod p, ( x), pn (x) being a polynomial of the ring R _{2 n} . 18. Device suitable for implementing the steps of the process according to the 20 previous claims. 19. Computer program comprising instructions adapted to the implementation of each of the steps of the method according to any one of claims 1 to 17 when said program is executed on a computer. 20. Information storage means, removable or not, partially or totally readable by a computer or a microprocessor comprising instructions of code of a computer program for the execution of each of the steps of the method according to any one of claims 1 to 17. 1/6 200,500 300 301 302 303 304 305 306