WO2004070956A1

WO2004070956A1 - Coding method and device

Info

Publication number: WO2004070956A1
Application number: PCT/IB2004/000873
Authority: WO
Inventors: Philippe Piret; Frédéric Lehobey
Original assignee: Canon Kabushiki Kaisha
Priority date: 2003-02-10
Filing date: 2004-02-06
Publication date: 2004-08-19
Also published as: WO2004070956A8; FR2851096A1

Abstract

In order to code an information sequence comprising k symbols, k being a positive integer, in the form of a code word of length n, n being an integer greater than k, this code word belonging to an algebraic geometry code, the information sequence of k symbols is supplemented with n-k redundancy symbols in order to form the code word, so that the k information symbols form a subset of the code word, using a plurality of coding operations by a ReedSolomon coder.

Description

CODING METHOD AND DEVICE

The present invention relates to a coding method and device.

The present invention concerns communication systems in which, in order to improve the fidelity of the transmission, the data to be transmitted are subjected to a channel coding.

So-called "channel" coding consists, when "code words" sent to the receiver are formed, of introducing a certain degree of redundancy in the data to be transmitted. More precisely, by means of each code word, a predetermined number k of information symbols chosen within a predetermined "alphabet" of finite dimension q are transmitted; from these k information symbols a number n of symbols taken from the same alphabet are formed, so as to form code words v = (vi, v₂, ... , v_n); the set of calculation rules for these n symbols according to the k information symbols defines a "code", or "coding method", of "dimension" k and "length" n, thus characterized by a certain set of code words constituting a kind of dictionary.

When the dimension q of the alphabet is a power of a prime number, it is possible to give to this alphabet a field structure, referred to as "Galois field F_q". In this case, some codes can, conveniently, be defined by means of a matrix H of size (n-k)χn defined on F_q, referred to as a parity-check matrix: a word v of given length n is a code word if, and only if, it satisfies the equation: H.v^τ = 0 (where the exponent T indicates the transposition).

Amongst known codes, the "Reed-Solomon codes" can be cited, which are renowned for their efficacy (for a definition of Reed-Solomon codes, reference can be made to the work by R.E. Blahut entitled "Theory and Practice of Error-Control Codes" , Addison-Wesley, Reading, Mass., 1983). These codes are defined on F_q.

However, Reed-Solomon codes have the particularity that the length n of the code words is necessarily less than or equal to the dimension q of the alphabet of the symbols. Because of this, if it is wished to have a Reed- Solomon code having code words of great length, it is necessary to envisage high values of q, which leads to expensive implementations with regard to the calculations and storage. In addition, high values of q are sometimes unsuited to the technical application envisaged.

However, in modern information media, for example in CD (Compact Disk) and DVD (Digital Video Disk) recordings, it is sought to increase the information density. When such a medium is affected by a physical defect such as a scratch, a large number of information symbols may be made illegible. It is however possible to remedy this problem by using a code of very great length. This is why it has been sought to construct codes offering in a natural manner a greater length of words than Reed-Solomon codes. In particular, codes known as "algebraic geometry codes" or geometric Goppa codes have recently been proposed (see for example "Algebraic Geometric Codes", by J.H. Van Lint, in "Coding Theory and Design Theory", Part 1 , IMA Volumes Math. Appl., Volume 21 , Springer- Verlag, Berlin, 1990). These codes, also defined on a Galois field F_q, are constructed from an algebraic equation with two unknowns X and Y. The solutions of this algebraic equation can be considered to be the coordinates (Xj.yi) of points on an "algebraic curve".

An algebraic geometry code C of length n and dimension k on the finite Galois field F_q can be described by means of a parity-check matrix H of size (n-k)χn and of rank n-k: C = {v s F_q ⁿ | H.v^τ = 0}, where ^τ designates the transposition, and where the matrix H is obtained as follows.

First of all, a curve is defined in F_q ² by means of an equation f(X,Y) = 0 where f(X,Y) is a bi-variant polynomial absolutely irreducible over F_q. The set of solutions (X = x, Y = y) of this equation is referred to as the "locating set". Hereinafter, it is assumed that the curve is a curve C_a, : the monomial of highest degree in X is X^a, the monomial of highest degree in Y is Y^b, possibly multiplied by a non-zero element of F_q, and (a,b) = 1 , that is to say a and b are relatively prime with each other. Next, the weight function ρ(X'Y^j) = bi + aj is associated with any monomial X'Y^j with i > 0 and b-1 > j > 0. For example, with f(X,Y) = Y⁴ + Y + X¹⁷, p(X'Y^J) = 4i + 17j. These monomials can be ordered according to the value of their weight function. In the example, the following Table 1 is obtained, where the monomials are ranged by increasing weight:

Table 1

Let {(X1N₁), ... , (x_n,Vn)} be the n solutions in F_q ² of f(X,Y) = 0. The first row of the matrix H contains the evaluation of the first monomial | (which is always equal to 1) at the n points of the locating set. Thus the first row in the matrix H is (11...11). The second row of H contains the evaluation of the second monomial f₂ (which is always equal to X) at the n points of the locating set. Thus the second row of H is (xi x₂ ... x_n-ι x_n). The construction of H continues in this way until the (n-k)^th row of H is reached. In the example in Table 1 , the 5^th row of H is the sequence of Xj⁴, the 6^th row of H is the sequence of yi and the 10^th row of H is the sequence of yjXj².

It is known that it is possible to code algebraic geometry codes whilst maintaining the complexity at a reasonable level. However, the majority of coders for algebraic geometry codes are not systematic. This means that the information symbols are not found again identically, that is to say not transformed, in the previously chosen positions of the coded word.

For example, the article by T. Yaghoobian and I.F. Blake entitled "Hermitian Codes as Generalized Reed-Solomon Codes", in Designs, Codes and Cryptography, Vol. 2, pp. 5-17, 1992, mentions the possibility of performing a simple coding of an algebraic geometry code by means of Reed-Solomon coders, but this coding does not have a systematic appearance in that the symbols to be coded are not found again identically in the coded sequence. There is also known the article by R. Matsumoto, M. Oishi and K.

Sakaniwa entitled "Fast Encoding of Algebraic Geometry Codes", which appeared in IEICE Trans. Fundamentals, Vol. E84-A, No. 10, pp. 2514-2517, October 2001 , but its purpose is not to obtain systematic coders.

Finally, the article by C. Heegard, J. Little and K. Saints entitled "Systematic Encoding via Grόbner Bases for a Class of Algebraic Geometric Goppa Codes" which appeared in IEEE Trans. On Inform. Theory, Vol. 41 , No. 6, pp. 1752-1761 , November 1995, proposes the construction of systematic coders, but using a method which depends on the construction of a Grδbner base, which is not the case with the present invention. The purpose of the present invention is to remedy the aforementioned drawbacks by providing a so-called systematic coding of algebraic geometry codes.

For this purpose, the present invention provides a method of coding an information sequence comprising k symbols, k being a positive integer, in the form of a code word of length n, n being an integer greater than k, this code word belonging to an algebraic geometry code, this method being notable in that it comprises a step consisting of supplementing the information sequence of k symbols with n-k redundant symbols in order to form the code word, so that the k information symbols form a subset of the code word, using a plurality of coding operations by a Reed-Solomon coder.

Thus, in each coded word, the information symbols appear unchanged at given locations and, by virtue of the Reed-Solomon codes, it is thus possible to code the algebraic geometry codes, systematically, with relative simplicity. The present invention can be applied equally well

- to an algebraic geometry code constructed on a curve f(X,Y) = 0, of type C_a,b, which contains bxq points (x,y) on a Galois field F_q with q elements, q being a power of a prime number and b being the highest degree in Y of the equation f(X,Y) = 0, or - to an algebraic geometry code constructed on a curve f(X,Y) = 0, of type C_a,b, which contains a number of points (x,y) strictly less than bxq on a Galois field F_q with q elements, q being a power of a prime number and b being the highest degree in Y of the equation f(X,Y) = 0, or

- to a code which, although not being an algebraic geometry code, has a parity-check matrix of one of the forms (1) or (13) given below, with t(0) > t(1) > ... and σ(0) 2 σ(1) 2 -.. In the first case, it is said that the curve f(X,Y) = 0 is attractive and in the second case it is non-attractive.

In a particular embodiment, the coding method according to the invention comprises a step consisting of writing any code word v, to be calculated from an item of information, in the form of a b-tuplet v = [v₀, ... , v_b-ι] of q-tuplets v^ in a finite Galois field F_q with q elements, b being a strictly positive integer, q being a power of a prime number and £ being an integer between 0 and b-1 , v satisfying the equation H.v^τ = 0, where H designates the parity-check matrix of this algebraic geometry code and ^T designates the transposition, and the coding method also comprises a step consisting of defining b words Uj of length q in F_q, i being an integer between 0 and b-1 , as follows:

where ^eϊ_e designates a diagonal matrix of size qxq having the element y_έ (γ^u_1) in position (u,u), u being an integer between 1 and q-1 , where γ is a primitive element of the field F_q and where the elements y₀(x), ... , yb-ι(x) are the solutions in Y of an equation f(x,Y) = 0 for any element x of the field F_q, and having the element y_£ (0) in position (q,q).

In another particular embodiment, the coding method according to the invention comprises a step consisting of writing any code word v, to be calculated from an item of information, in the form of a b-tuplet v = [vo, ... , n] of words v i of length n^ in a finite Galois field F_q with q elements, I being an integer between 0 and b-1 , the n^ values satisfying n^ ≥ _£+^ , b being a strictly positive integer, q being a power of a prime number and v satisfying the equation H.v^τ = 0, where H designates the parity-check matrix of said algebraic geometry code and ^τ designates the transposition, and the method also comprises: - a step consisting of supplementing each word v_e with n₀ - n^ zeros, and then

- a step consisting of defining b words Uj of length n₀ in F_q, i being an integer between 0 and b-1 , as follows: u₀ = v₀ +. . + v b-1 ' v₀ ^<r₀ +... +v_b_₁r_t b-1 '

(3) b-1

I" b-1 = V₀*o + ... + V, yb-1 -I b-1 ^• where ¥_e designates a diagonal matrix of size no no where the first n^ diagonal elements contain the (^ +1)^th solutions y^(x) associated with the n_e values of x for which the equation f(x,Y) = 0 admits at least £ +1 solutions in Y.

The above two particular embodiments make it possible to associate, with the code word v of length b.q, which is to be calculated from an item of information, b words Uj (cf. equations (3)) which are easier to calculate than the word v.

According to a particular feature, the method comprises a step consisting of rewriting the equation H.v^τ = 0 in the form of a set of b equations, as follows:

where H_t designates a matrix of size txq, t being a positive integer, having the element γ^(l"1)(i"1) in position (i,j) for any integer i between 1 and t and for any integer j between 1 and q-1 , having the element 1 in position (1,q) and having 0 in the other positions in the last column and where, for any integer m between 0 and b-1 , t(m) designates the number of rows in the matrix Ht(_m)-

According to another particular feature, the method according to the invention comprises a step consisting of rewriting the equation H.v^τ = 0 in the form of a set of b equations, as follows:

where, for 1 <

n , t(£) being a positive integer, having in position (i,j), for 1 < i < t(£) and 1 < j < n_£ , the element

γ('- )(j-l) _{( w} ere is the G+1)^th of the elements x of F_q, for which the equation

f(x,Y) = 0 admits at least £ +λ solutions in Y.

There is thus obtained, by virtue of equations (4), a principle for determining the b words uι of length q.

According to another particular feature, the coding method comprises b calculation steps for obtaining the words Uι for any integer i between 0 and b- 1 , the s^th calculation step, for any integer s between 1 and b, using the s^th equation of the system of equations (4) for calculating the last t(s-1) symbols of u_s-ι by means of a Reed-Solomon coder.

This constitutes a practical method for implementing the principle of determining the Uj as briefly described above.

The present invention does not apply only to algebraic geometry codes. Thus, for the same purpose as mentioned above, the present invention also provides a method for the systematic coding of an information sequence comprising k symbols on a finite Galois field F_q with q elements, q being a power of a prime number and k being a positive integer, in the form of a code word of length n on F_q, n being an integer greater than k, characterized in that this code word belongs to a code having as a parity-check matrix a matrix of the form:

H_t(o) "t(0) H_t(0)

' 't(1)^r ) H ⁿt(1) <-Y^t 1 ' 't(1)^f >-1

H = H ^πt(₂) Υ ^JL o² H ^πt(₂) <y ^z1^{2 π}t(₂) -^£ b-1 (1)

_ H^πt(b-1) ^x 0^{0-1 n}t(b-1) ²1 ^πt(b-1) ² b-1 where H designates a matrix of size txq, t is a positive integer, the element in position (i,j) of H is γ^(l"1)(i"1) for any integer i between 1 and t and for any integer j between 1 and q-1 , the element in position (1 ,q) of H is 1 , the element in position (i,q) of H is 0 for i>0, γ is a primitive element of F_q, the t(m) values satisfying t(m) > t(m+1) for m = 0, ... , b-2 and ^<γ_t is, for any integer £ between 0 and b-1 , a diagonal matrix of size qxq having the element y^_u in position (u,u), these elements y_{e u} all being different for a fixed £ .

This coding method has the same advantages as the coding method succinctly disclosed above. Still for the same purpose, the present invention also provides a method for the systematic coding of an information sequence comprising k symbols on a finite Galois field F_q with q elements, q being a power of a prime number and k being a positive integer, in the form of a code word of length n on F_q, n being an integer greater than k, notable in that this code word belongs to a code having as a parity-check matrix a matrix of the form:

H = (13)

where the σ(f) are subsets of F_q satisfying σ(f) 3 σ(f+1) for f = 0, ... , b-2, H comprises t rows and a number of columns equal to the number of elements in σ, the element in position (i,i) of H is x ^"1 where X_j represents the j^th element of σ, 0° represents 1 and the elements of the matrices ^<Y £ = 0, ... , b-1 , which are associated with the same Xj are all different.

This coding method has the same advantages as the coding method succinctly disclosed above.

Still for the same purpose, the present invention also provides a device for coding an information sequence comprising k symbols, k being a positive integer, in the form of a code word of length n, n being an integer greater than k, this code word belonging to an algebraic geometry code, this device being notable in that it comprises means for supplementing the information sequence of k symbols with n-k redundant symbols in order to form the code word, so that the k information symbols form a subset of the code word, using a plurality of coding operations by a Reed-Solomon coder.

Still for the same purpose, the present invention also provides a device for the systematic coding of an information sequence comprising k symbols on a finite Galois field F_q with q elements, q being a power of a prime number and k being a positive integer, in the form of a code word of length n on F_q, n being an integer greater than k, notable in that this code word belongs to a code having as a parity-check matrix a matrix of the form:

H_t(o) H_t(o) " )

•~'t(1)'^l 0 H_j^ , ^■~'t(1)'^I b-1

H = H ^πt(2) <r ^z 0² H ^πt(2) Υ ^z 1² ■ • H ^πt(2) < ^ZYb²-1 (1)

_ H^πt(b-l) ^z 0^0-1 ¹ 't(b-1) ^z1 ⁿt(b-1) ^zb-1 _ where Ht designates a matrix of size txq, t is a positive integer, the element in position (i,j) of H is γ^(l"1)(i"1) for any integer i between 1 and t and for any integer j between 1 and q-1 , the element in position (1 ,q) of H is 1 , the element in position (i,q) of H is 0 for i>0, γ is a primitive element of F_q, the t(m) satisfy t(m) > t(m+1) for m = 0, ... , b-2 and nf_i is, for any integer £ between 0 and b-1 , a diagonal matrix of size qxq having the element y_{i u} in position (u,u), these elements y_{i u} all being different for a fixed £ .

H

-1 (13) μσ(0) _cγ b-1 μσ(1) ry - μσ b-I -l ⁿt(b-1) ^z 0 ^πt(b-1) ^z1 ⁿt(b-1) ^z b-1 where the σ(f) are subsets of F_q satisfying σ(f) 3 σ(f+1) for f = 0, ... , b-2, H^ comprises t rows and a number of columns equal to the number of elements in σ, the element in position (i,i) of H° is x ^"1 where Xj represents the j^th element of σ, 0° represents 1 and the elements of the matrices ^<Y_t , £ = 0, ... , b-1 which are associated with the same Xj are all different.

The present invention also relates to a digital signal processing apparatus comprising means adapted to implement a coding method as above. The present invention also relates to a digital signal processing apparatus comprising a coding device as above. The present invention also relates to a telecommunications network comprising means adapted to implement a coding method as above.

The present invention also relates to a telecommunications network comprising a coding device as above.

The present invention also relates to a mobile station in a telecommunications network, comprising means adapted to implement a coding method as above.

The present invention also relates to a mobile station in a telecommunications network, comprising a coding device as above.

The present invention also relates to a base station in a telecommunications network, comprising means adapted to implement a coding method as above.

The present invention also relates to a base station in a telecommunications network, comprising a coding device as above.

The invention also relates to an information storage medium which can be read by a computer or a microprocessor storing instructions of a computer program for implementing a coding method as above.

In a particular embodiment, the information storage medium according to the present invention is partially or totally removable.

The invention also relates to a computer program product comprising sequences of instructions for implementing a coding method as above. The particular features and the advantages of the coding devices, of the various digital signal processing apparatus, of the various telecommunications networks, of the various mobile stations, of the various base stations, of the various storage medium and of the computer program product being similar to those of the coding method according to the invention, they are not repeated here.

Other aspects and advantages of the invention will emerge from a reading of the following detailed description of particular embodiments, given by way of non-limiting example. The description refers to the accompanying drawings, in which:

- Figure 1 illustrates schematically the format of the coding according to the present invention, in a particular embodiment, for the curve Y⁴ + Y + X¹⁷ = 0;

- Figure 2 is a flow diagram illustrating the b steps of the coding according to the present invention, in a particular embodiment;

- Figure 3 depicts the structure of the symbols of the Galois field F₁₆ for the codes according to the present invention associated with the curve Y³ + Y + X⁴ + X = 0;

- Figure 4 depicts, in a simplified schematic form, a telecommunications network according to the present invention; and

- Figure 5 illustrates schematically the constitution of a network station or computer transmission station adapted to implement a coding method according to the present invention.

Hereinafter, coders according to the present invention are first of all described in a particular embodiment limited to the context of a so-called attractive curve.

Assume that the degree b in Y of f(X,Y) is greater than or equal to 2. The curve defined by f(X,Y) = 0 is said to be attractive on a Galois field F_q with q elements, q being a power of a prime number, if it contains exactly bxq points (x,y) on F_q.

It is easy to verify that the curve defined by Y⁴ + Y + X¹⁷ = 0 is an attractive curve on F₂5₆ since, for any x e F₂₅β, Y⁴ + Y + x¹⁷ = 0 has exactly four distinct roots in F₂₅₆. More generally, the curve defined by f(X,Y) = 0 is attractive on F_q when, for any x e F_q, f(x,Y) = 0 has b distinct solutions in Y. They are noted yo(x), ... , yb-ι(x) and the locating set is reordered as:

(^χι.yo(^χι)) (X2,y₀(^χ ₂)) •■• (^χ _q,yo(χ_q)) (^χι,yι(^χ _q)) ■■• (^χ _qιyι(^χ _q)) ■■■ (^χ _q,y_b- (^χ _q))-

In addition, for a primitive element γ of F_q (that is to say where the first q-1 powers enumerate the non-zero q-1 elements of F_q), the q elements Xj of F_q are ordered in the form xi = 1 , x₂ = γ, ... , x_q-ι = γ^q"2, x_q = 0. The order of the rows in the parity-check matrix H is also modified. Instead of constructing its rows by successive evaluation of the monomials X'Y^j arranged in accordance with the increasing values of p, the matrix is constructed by juxtaposition of b matrices H® for j = 0, ... , b-1 , where H® designates all the rows of H associated with the monomials X'Y^j ordered in accordance with the increasing values of i, i = 0, ... , t(j)-1. Ht is also defined as being the matrix of size txq having γ^(l"1)(i"1) in position (i,j) for 1 < j < q-1 , having 1 in position (1 ,q) and having 0 in the other positions in the last column. By way of non-limiting example, the matrix Ht for

1 1 1 1

14

Y γ¹³ Y

H ..2 ..11 ..13

Y

1 γ^ 12 Y 1 _γ ^< .11

with γ 1¹5⁰ = 1

Finally, for £ = 0, ... , b-1 , ^<Y_t can be defined as being the diagonal matrix of size qxq having y^ (γ^u~1) in position (u,u) for u = 1 , ... , q-1 and y^ (0) in position (q,q).

By taking account simultaneously of all these notations, the matrix H is rewritten in the form:

H (1)

In the particular case of the curve Y⁴ + Y + X¹⁷ = 0 on F₂₅₆, if all the monomials X'Y^J are chosen such that 4i + 17j < 123, exactly 100 monomials of this type are obtained and the numbers t(j) are t(0) = 31 , t(1) = 27, t(2) = 23 and t(3) = 19. The matrix H is then given by:

^HS1 ^H31 ^H31 ^H31

H₂₇r₀ H₂₇^ H_{27 2} HΛ

H = (2)

H ^π23 ^<Y ^Z 0² H π₂₃ ^<Y ι₁ ² H ⁿ23 T ^Z 2² '~'23^fz 3

. H19 ry^z-0³ H ' '19 ^z1³ H ' ^!19 <Y^z" ₂ ³ H19 Υ^z 3^z

Next comes the description of the invention proper. It is assumed that k information symbols are available and that n-k symbols are to be appended following the sequence formed by the k information symbols, in order to form a code word of length n in an algebraic geometry code associated with an attractive curve. Thus it is wished for the information symbols to appear unchanged somewhere amongst the n coded symbols. This type of coding is called a systematic coding.

To do this, any code word v, to be calculated from an item of information, is written in the form of a b-tuplet v = [v₀, ... , V -i] of q-tuplets v_έ , £

= 0, ... , b-1 on F_q. In order to be a code word, v must satisfy H.v^τ = 0. Thus, in order to construct v from a k-tuplet of information symbols, b words Uj of length q on F_q are defined by: u₀ = v₀ +... + v_b_₁,

U₁ = V₀ ^<V₀ + ... + V_b_₁Υ_b.₁,

(3) u_b_₁ = v₀r .₀b°--1^, +...+v_b_₁'r -_bb^u-1

This results in reformulating the equation H.v^τ = 0 in the form of a set of b equations:

The q-tuplets Vj and Uj are written in the form of concatenations of b+1 pieces: v_i = [v_i ⁰, Vi¹, ... , v_i ^b], Ui = [Ui⁰, Ui¹, ... , Ui^b]. (5)

In the definitions (5), the length of v,⁰ and u ° is q-t(0), ... , the length of v^¹ and u^¹ is t(j)-t0+1), ... , the length of V|^b and Uj^b is t(b-1).

In our example, the four words Vj have the length 256, the four pieces Vj° have the length 225, the four pieces v⁴ have the length 19 and the other pieces v} have the length 4. This is depicted in summary form in Figure 1 , which gives the format of the coding for the curve Y⁴ + Y + X¹⁷ = 0: the pieces Vo°, Vι°, v₂°, v₃°, V- v₂ ¹, v₃ ¹, v₂ ², v₃ ² and v₃ ³ contain the information symbols and the other pieces contain redundant symbols.

Consider once again the case of a code on an attractive curve for calculating the parity-check symbols.

First of all, from the system of equations (3), Uo is given by:

Thus the first q-t(0) symbols of Uo are known, knowing that the b pieces Vj°, i = 0, ... , b-1 are information pieces. Use will now be made of the first equation of the system (4):

Ht₍₀₎u₀ ^T = 0 (7) which implies that it is possible to calculate the last t(0) symbols of u₀ (that is to say [uo¹, ... , Uo ]) by means of a Reed-Solomon coder. Thus, at this time, the information pieces Vj¹ for i = 1 , ... , b-1 are known, and also Uo¹ = v₀ ¹ + vi¹ + ... + V_b-ι¹ is known. This is sufficient for calculating Vo¹. Note that the pieces Uo², ... , Uo^b which will be used subsequently for calculating other parity pieces are also known by virtue of this step.

From the system of equations (3), ui is given by: uι = v₀% + ... + v_b-ι%,₁ (8)

Thus the first q-t(1) symbols of ui are known, knowing that the elements Vj°, i = 0, ... , b-1 and Vj¹, i = 1 , ... , b-1 are pieces of information and knowing that the element Vo¹ was calculated during the previous step. Use will be made now of the second equation of the system (4):

which implies that it is possible to calculate the last t(1) symbols of Ui (that is to say [u-i², ... , uι^b]) by means of a Reed-Solomon coder. Thus, at this time, the information pieces Vj¹ for i = 1 , ... , b-1 and v² for i = 2, ... , b-1 are known, and Uo¹ is also known by virtue of the previous step and u-i² by virtue of the present calculation. By means of the first two (linear) equations of system (3), sufficient information is available for calculating v₀ ² and vi². It should also be noted that, by virtue of this step, the pieces ui³, ... , Uι^b which will be used subsequently for calculating other parity pieces are also known.

It is possible to continue in this way to successively calculate u₂, ... , U_b-i. At the s^th step, s = 1 , ... , b, the s^th equation of system (4) is used for calculating the last t(s) symbols of u_s by means of a conventional Reed- Solomon coder. Thus, knowing U ^S, i = 1 , ... , s-1 by virtue of the previous steps and u_s-ι^s by virtue of the last step, it is possible to use the first s equations of the system (3) for calculating Vo^s, ... , v_s-ι^s. The coding process can be resumed as illustrated by Figure 2. After obtaining the numbers t(0), t(1), ... , t(b-1) at step 20, the b words Uj of length q (given by equation (3)) at step 22 are defined. Then, at step 24, from the first equation of the system (4), the b-tuplet [uo¹, ... , Uo ] is obtained, and, from the first row of the system (3), the piece v₀ ¹ is obtained. At the following step 26, from the second equation in the system (4), the (b-l)-tuplet [u-i², ... , u-ι^b] is obtained, and from the first two rows of the system (3) the pieces v₀ ², v-i² are obtained, and so on as far as step 28, during which, from the last equation in the system (4), the singleton [U_b-ι^b] is obtained, and from all the equations in the system (3) the pieces vo^b, ..., v_b-ι^b are obtained. The way in which the present invention is extended to coders on curves which are not necessarily attractive is described below.

Stating that a curve C_a,b with b < a is not attractive on F_q amounts to stating that it has a number of points strictly less than b.q on F_q. This means that, in a parity-check matrix H such as the one given at (2), some columns are to be omitted. By way of in no way limiting example, the curve defined on F-i₆ by Y³ + Y + X⁴ + X = 0 is considered. It has 32 finite points (x,y) which are listed in Table 2, where γ is a root of X⁴ + X + 1.

Table 2

This curve a has a genus g = 3. It also has a weighting function for the monomials, which is given by p(X'Y^J) = 3i + 4j. For this example, the Table 3 of weights below is obtained:

Table 3

Let the 32 solutions in F₁₆ of Y³ + Y + X⁴ + X = 0 be {(xι,yι), ..., (X₃2,Y₃2)}- A parity-check matrix H is constructed for an algebraic geometry code of length n = 32 and of dimension k = 21. The minimum distance d of this code will satisfy d > n-k+1-g = 9. The first row of H contains the evaluation of the first monomial fi = 1 at the 32 points of the locating set. The second row of H contains the evaluation of the second monomial f₂ = X at the 32 points of the locating set. Thus the first row of H is (xi x₂ ... x_n-ι x_n). The construction of H continues in this way until the 11^th row of H is obtained. In the example in Table 3, the 5^th row of H is the sequence of yjXj, the 6^th row of H is the sequence of y² and the 10^th row of H is the sequence of x⁴.

Next the order of the rows in this parity-check matrix is modified. Instead of constructing its rows by successive evaluation of the monomials X'Y^j ordered in accordance with the increasing values of p, it is constructed as the juxtaposition of b = 3 matrices H® for j = 0, 1 , 2, where H® designates the set of rows of H associated with the monomials X'Y^j ordered in accordance with the increasing values of i, i = 0, ..., t(j)-1. In the case of our example, t(0) = 5, t(1) = 4 and t(2) = 3.

An advantageous feature of the codes on an attractive curved defined by f(X,Y) = 0 is that, for any x e F_q, there are exactly b different values of Y in F_q which satisfy f(x,Y) = 0. This property is not satisfied in the above example so that it is no longer possible to give to H the advantageous structure given at (1), an example of which is given at (2). Thus it is necessary to continue the analysis of this more general situation.

For 8 values x of X, that is to say for x e {γ³, γ⁶, γ¹², γ⁹, γ⁷, γ¹⁴, γ¹³, γ¹¹}, there are 3 solutions in F-ι₆ for Y³ + Y + x⁴ + x = 0; for 4 values x of X, that is to say for x e {0, 1 , γ⁵, γ¹⁰}, there are 2 solutions in F-|₆ for Y³ + Y + x⁴ + x = 0; and for 4 values x of X, that is to say for x e {γ, γ², γ⁴, γ⁸}, there is no solution of Fι₆ for Y³ + Y + x⁴ + x = 0.

Given that there is a set σ(0) of 12 elements Xj, i = 1 , ... , 12, in F-i₆ for which there exists at least one solution Y = yo(Xi) in F-ι₆ for Y³ + Y + x⁴ + Xj = 0, the matrix H₅ ^σ(0) of size 5 x 12 which has Xj^1"1 as input (j,i) is constructed. Also % is defined as being the diagonal matrix of size 12 x 12 having yo(Xu) as input (u,u).

Given that there is a set σ(1) = σ(0) of 12 pieces Xj, i = 1 , ... , 12, in Fie for which there exists at least one second solution Y = y-_t(Xj) in F ₆ for Y³ + Y + Xj⁴ + Xj = 0, the matrix H₅ ^σ(1) of size 5x12 which has \^Λ as input (j,i) is constructed. Also is defined as being the diagonal matrix of size 12x12 having yι(x_u) as input (u,u). Finally, given that there is a set σ(2) of 8 pieces x-,, i = 1 , ... , 8, in F₁₆ for which there exists a third solution Y = y₂(Xj) in Fι₆ for Y³ + Y + x⁴ + Xj = 0, the matrix H₅ ^σ(2) of size 5x8 which has xf^1"1 as input (j,i) is constructed. Also Υ₂ is defined as being the diagonal matrix of size 8x8 having y₂(x_u) as input (u,u).

Taking all these notations again, the matrix H is rewritten in the form:

H = μ π^σ ₄ ⁽°⁾fy ι ₀ f> r₂ (11) ^π3 ^Z 0 H ² H >r₂ ²

Any code word v is written the form of a triplet v = [v₀, vi, v₂] where v₀ and vi are 12-tuplets and v₂ is an 8-tuplet. It should also be noted that the sets σ(i) satisfy σ(2) e σ(1) c σ(0). Thus it is possible to proceed as follows. First of all the list L₂ of the 8 pieces Xj in σ(2) is constructed. The list, which becomes a list Li of 12 pieces, is supplemented by appending to Li the four pieces Xj which are in σ(1) but not in σ(2). Finally it would also be possible to supplement Li in order to convert it into an extended list Lo by the elements which are in σ(0) but not in σ(1). However, in the specific case considered, the two sets σ(1) and σ(0) are identical. The lists Li and L₀ are therefore also identical.

On the basis of the elements which have just been introduced, the coding of a code on Fi₆ defined on the non-attracting curve Y³ + Y + X⁴ + X = 0 can be carried out as follows. Rather than defining the parity-check matrix of the code by a matrix H as given at (11), it should be attempted to define it by a matrix H as given at (1). In the example in question, the number of rows in this matrix is t(0) + t(1) + t(2) = 12, which is indeed the number of parity equations to be satisfied by the words of the code.

Three difficulties arise, however, to which it will be necessary to respond. - The number of columns in the matrix H obtained according to (1) is 16 + 16 + 16 = 48. This number therefore exceeds the length of our code, which is 32, by 16 units. - Some elements of the matrices % are indeterminate since, for certain values x of X, the equation Y³ + Y + x⁴ + x = 0 admits less than three solutions in Y.

The third difficulty will be stated after having explained how to resolve the first two.

The first difficulty is resolved as follows. If suffices to force 16 symbols v} to be zero. These 16 symbols forced to be zero will not be transmitted, and the true length of the code will therefore be equal to 48-16 =

32. These 16 symbols v forced to be zero will be those for which i and j are such that γ^1' does not belong to σ(i).

The second difficulty is resolved as follows. In the example in question, equation (3) is written:

(12)

where some components of the % values can be indeterminate. It should be noted that the indeterminate components are always, in equation (12), allocated as coefficients of symbols v which are forced to be zero. Consequently these components can remain indeterminate since, in fact, they "do not appear" in equation (12).

The third difficulty announced above is now set out. - It must be possible to be able to find, as redundant symbols to be calculated, symbols Vf¹ which are not forced to be zero and to which it is possible to apply the calculation algorithm described above during the description of the coding of the codes defined on attractive curves.

This third difficulty is resolved as follows. It is known that, in a Reed- Solomon code of length n and dimension k, any set of k symbols can be taken as an information set and therefore also any set of n-k symbols can be taken as a redundant set. This property is a consequence of the "Maximum Distance Separable" (or MDS) property of these codes. In this regard reference can usefully be made to the work by F.J. MacWilliams and N.J.A. Sloane entitled "The Theory of Error-Correcting Codes", published by North-Holland in 1977. The coding procedure explained above will therefore be applicable provided that there exist, in σ(0), σ(1) and σ(2), subsets σ'(0) of cardinal t(0), σ'(1) of cardinal t(1) and σ'(2) of cardinal t(2) which are such that σ'(0) 3 σ'(1) 3 σ'(2). In the example in question, these subsets are σ'(0) = {γ¹¹, γ¹², γ¹³, γ¹⁴, γ^∞}, σ'(1) = {γ¹², γ¹³, γ¹⁴, γ^∞} and σ'(2) = {γ¹³, γ¹⁴, f}.

On the basis of the above discussion, the coding can take place in the same way for a code on a non-attractive curve as for a code on an attractive curve, provided that the condition σ'(0) 3 σ'(1) 3 σ'(2) 3 ... is satisfied.

The coding of the codes on F₁₆, of length 32, redundancy 12 and orthogonal to the matrix (11) is specified by Figure 3. In the drawing, above the table, the indices j which appear are the integer numbers from 0 to 14 as well as the symbol 00. These numbers and this symbol are the image of the 16 elements of Fι₆ which appear as the second line of the matrices Ht. The number j is the image of γ", where γ is a primitive element of F-ι₆ and the symbol co is the image of the zero of F₁₆. In the three framed lines in the table in Figure 3 there are depicted the symbols vi which can be different from zero. If the corresponding symbol

is forced to be zero, it is simply written as 0.

In this example, it has been seen that it was possible to choose as sets σ'(i) the elements of Fι₆ associated with the last t(i) elements of each of the framed lines in the table in Figure 3. This choice is however not obligatory. It is for example possible to choose these sets σ'(i) as follows: σ'(0) = {γ¹⁰, γ⁹, γ⁷, γ¹⁴, γ¹³}, σ'(1) = {γ⁹, γ⁷, γ¹⁴, γ¹³} and σ'(2) = {γ⁹, γ⁷, γ¹³}. The only condition to be fulfilled is that these sets are nested (σ'(0) 3 σ'(1) 3 σ'(2)) and that they do not make reference to pieces v^ forced to be zero. The nullity of some of the components v of the code word is imposed so that this word, after elimination of these zero components, will be orthogonal to the corresponding matrix of type (11). The coding can then take place like that of an attractive curve. In particular:

- the symbols of the first 11 columns (numbered 0 to 10) in Figure 3 are information symbols and are therefore known. The first 11 symbols of u₀, are therefore known and it is possible to calculate the last 5 symbols of uo using H₅.Uo^τ = 0; - already knowing v-i¹¹ and v₂ ¹¹, which are information symbols, it is therefore possible to calculate v₀ ¹¹ = Uo¹¹ - Vi¹¹ - v₂ ¹¹;

- the symbols of the first 12 columns in Figure 3 are now known. The first 12 symbols of ui are therefore known and it is possible to calculate the last 4 symbols of Ui using H₄.u₀ ^T = 0;

- knowing the thirteenth symbol Uo and of Ui, it is now possible to calculate the symbols v₀ ¹² and vi¹², and so on.

In the general case of a non-attractive curve C_a,b, the parity-check matrix H will have the following form: μσ(0) uσ(1) μσ(b-1) ^πt(0) ⁿt(0) ⁿt(0) μ^σ(°)ry μσ(i)<y μσ(b-1)_ry-

H = ^πt(1) ^z o ^πt(1) ^z 1 ^πt(1) ^z b- (13) μσ(0) ,yb-1 μσ(1) ^y-b-1 ua(b-1)_fyb-1

_^πt(b-1) ^z u ^πt(b-1) ^z 1 ⁿt(b-1) b-1 _ where each set σ(j) is either the set of the q elements Xj over F_q, or one of its subsets. These subsets σ(j) satisfy σ(j+1) Q σ(j) for j = 0, ... , b-2, and the matrix orj is the diagonal matrix of the yj(Xj) values for Xj in σ(j). It is said that ^<y] is associated with σ(j). The coding method as described above and in particular the coding method for codes defined on attractive curves can be extended to codes which are not algebraic geometry codes and which are therefore not associated with curves. It suffices for this purpose for their parity-check matrix to have the form (1) or (13) with t(0) > t(1) > ... and σ(0) 3 σ(1) 3 ... and that, for any given integer i, the elements in position (i,i) in the matrices <Y_t are all different. In this more general case, the same coding algorithm as that described above can in fact be used, even if the minimum distance of the code can no longer be limited at the bottom by algebraic geometry arguments.

As shown by Figure 4, a telecommunications network according to the invention consists of at least one station known as a base station SB designated by the reference 64 and several peripheral stations known as mobile terminals SPi, i = 1 M, where M is an integer greater than or equal to 1 , respectively designated by the references 66₁, 66₂, ... , 66M. The peripheral stations 66₁, 66₂, ... , 66_M are distant from the base station SB, each connected by a radio link with the base station SB and able to move with respect to the latter.

The base station 64 can comprise means adapted to implement a coding method according to the invention. In a variant, the base station 64 can comprise a coding device according to the invention. Similarly, at least one of the mobile terminals 66j can comprise means adapted to implement a coding method according to the invention or comprise a coding device according to the invention.

In the particular embodiment in Figure 4, the invention applies to a wireless network. Nevertheless the invention is adapted to be applied to any communication system, independently of the cabled or wireless character of the telecommunications network.

Figure 5 illustrates schematically the constitution of a network station or computer transmission station, in the form of a block diagram. This station comprises a keyboard 911 , a screen 909, an external information source 910, and a radio transmitter 906, conjointly connected to an input/output port 903 of a processing card 901.

The processing card 901 comprises, connected together by an address and data bus 902: - a central processing unit 900;

- a random access memory RAM 904;

- a read only memory ROM 905; and

- the input/output port 903.

Each of the elements illustrated in Figure 5 is well known to persons skilled in the art of microcomputers and transmission systems and, more generally, information processing systems. These common elements are therefore not described here. It should however be noted that:

- the information source 910 is for example an interface peripheral, a sensor, a demodulator, an external memory or other information processing system (not shown), and is preferably adapted to supply sequences of signals representing speech, service messages or multimedia data, in the form of sequences of binary data, and that - the radio transmitter 906 is adapted to implement a packet transmission protocol on a non-cabled channel, and to transmit these packets over such a channel.

It should also be noted that the word "register" used in the description designates, in each of the memories 904 and 905, both a memory area of small capacity (a few binary data) and a memory area of large capacity (for storing an entire program).

The random access memory 904 stores data, variables and intermediate processing results in memory registers bearing, in the description, the same names as the data whose values they store. The random access memory 904 comprises in particular:

- a register "data_to_be_coded in which the digital signals to be coded are stored.

The read only memory 905 is adapted to store, in registers which, for convenience, have the same names as the data which they store:

- the operating program of the central processing unit 900, in a register "program".

The central processing unit 900 is adapted to implement a coding method as illustrated by the flow diagram in Figure 2.

Claims

1. Method of coding an information sequence comprising k symbols, k being a positive integer, in the form of a code word of length n, n being an integer greater than k, said code word belonging to an algebraic geometry code, said method being characterized in that it comprises a step consisting of supplementing said information sequence of k symbols with n-k redundancy symbols in order to form said code word, so that the k information symbols form a subset of the code word, using a plurality of coding operations by a Reed- Solomon coder.

2. Method according to Claim 1 , characterized in that said algebraic geometry code is constructed on a curve f(X,Y) = 0 which contains bxq points (x,y) on a Galois field F_q with q elements, q being a power of a prime number and b being the highest degree in Y of the equation f(X,Y) = 0.

3. Method according to Claim 1 , characterized in that said algebraic geometry code is constructed on a curve f(X,Y) = 0 which contains a number of points (x,y) strictly less than bxq on a Galois field F_q with q elements, q being a power of a prime number and b being the highest degree in Y of the equation f(X,Y) = 0.

4. Method according to Claim 1 or 2, characterized in that it comprises a step consisting of writing any code word v, to be calculated from an item of information, in the form of a b-tuplet v = [v₀, ... , V -ι] of q-tuplets

in a finite Galois field F_q with q elements, b being a strictly positive integer, q being a power of a prime number and £ being an integer between 0 and b-1 , v satisfying the equation H.v^τ = 0, where H designates the parity-check matrix of said algebraic geometry code and ^τ designates the transposition, and in that said method also comprises a step consisting of defining b words Uj of length q in F_q, i being an integer between 0 and b-1 , as follows: u₀ = v₀ + ... + v_b__{1 l} u₁ = v₀r₀ +... +v_b_₁τr_b._{1 l}

u_b_ι = v₀ -¹ +...+v_b_₁r_b ^b-¹. where Υ_e designates a diagonal matrix of size qxq having a position (u,u), u being an integer between 1 and q-1 , the element y_£ (γ^u"1), where γ is a primitive element of the field F_q and where the elements yo(x), ... , yb-i(x) are the solutions in Y of an equation f(x,Y) = 0 for any element x of the field F_q, and having the element y^ (0) in position (q,q).

5. Method according to Claim 1 or 3, characterized in that it comprises a step consisting of writing any code word v, to be calculated from an item of information, in the form of a b-tuplet v = [v₀, ... , v_b-ι] of words v^ of length _e in a finite Galois field F_q with q elements, £ being an integer between 0 and b-1 , the _£ satisfying n^ > n^₊₁ , b being a strictly positive integer, q being a power of a prime number and v satisfying the equation H.v^τ = 0, where H designates the parity-check matrix of said algebraic geometry code and ^τ designates the transposition, and in that said method also comprises:

- a step consisting of supplementing each word v^ with nr_j - n^ zeros, and then

- a step consisting of defining b words Uj of length no in F_q, i being an integer between 0 and b-1 , as follows:

where ^<Y_t designates a diagonal matrix of size noχn₀ where the first n^ diagonal elements contain the (^ +1)^th solutions y^(x) associated with the n_£ values of x for which the equation f(x,Y) = 0 admits at least £ +1 solutions in Y.

6. Method according to Claim 4, characterized in that it comprises a step consisting of rewriting the equation H.v^τ = 0 in the form of a set of b equations, as follows:

where Ht designates a matrix of size txq, t being a positive integer, having the element γ^(l_1)(i"1) in position (i,j) for any integer i between 1 and t and for any integer j between 1 and q-1 , having the element 1 in position (1,q) and having 0 in the other positions in the last column and where, for any integer m between 0 and b-1 , t(m) designates the number of rows in the matrix H_t(m)-

7. Method according to Claim 5, characterized in that it comprises a step consisting of rewriting the equation H.v^τ = 0 in the form of a set of b equations, as follows:

where, for

, t(£) being a positive integer, having in position (i,j), for 1 < i < t(^) and 1 < j < n^ , the element

γ(ι- )(M) _f where γ_j is the (j+1)^th of the elements x of F_q, for which the equation

f(x,Y) = 0 admits at least £ + solutions in Y.

8. Method according to Claim 6 or 7, characterized in that it comprises b calculation steps for obtaining the words Uj for any integer i between 0 and b-1 , the s^th calculation step, for any integer s between 1 and b, using the s^th equation of the system of equations (4) for calculating the last t(s- 1) symbols of u_s-ι by means of a Reed-Solomon coder.

9. Method for the systematic coding of an information sequence comprising k symbols on a finite Galois field F_q with q elements, q being a power of a prime number and k being a positive integer, in the form of a code word of length n on F_q, n being an integer greater than k, characterized in that said code word belongs to a code having as a parity-check matrix a matrix of the form:

H (1)

where H_t designates a matrix of size txq, t is a positive integer, the element in position (i,j) of H is γ^(l"1)(i"1) for any integer i between 1 and t and for any integer j between 1 and q-1, the element in position (1 ,q) of H is 1 , the element in position (i,q) of H is 0 for i>0, γ is a primitive element of F_q, the t(m) satisfy t(m) > t(m+1) for m = 0, ... , b-2 and ^cϊ_l is, for any integer £ between 0 and b-1 , a diagonal matrix of size qxq having the element y_{i u} in position (u,u), these elements y_{e u} all being different for a fixed £ .

10. Method for the systematic coding of an information sequence comprising k symbols on a finite Galois field F_q with q elements, q being a power of a prime number and k being a positive integer, in the form of a code word of length n on F_q, n being an integer greater than k, characterized in that said code word belongs to a code having as a parity-check matrix a matrix of the form:

where the σ(f) are subsets of F_q satisfying σ(f) 3 σ(f+1) for f = 0, ... , b-2, H^ comprises t rows and a number of columns equal to the number of elements in σ, the element in position (i,i) of H^ is X_j'^"1 where Xj represents the j^th element of σ, 0° represents 1 and the elements of the matrices <Y_e , £ = 0, ... , b-1 , which are associated with the same x, are all different.

11. Device for coding an information sequence comprising k symbols, k being a positive integer, in the form of a code word of length n, n being an integer greater than k, said code word belonging to an algebraic geometry code, said device being characterized in that it comprises means for supplementing said information sequence of k symbols with n-k redundancy symbols in order to form said code word, so that the k information symbols form a subset of the code word, using a plurality of coding operations by a Reed- Solomon coder.

12. Device according to Claim 11 , characterized in that said algebraic geometry code is constructed on a curve f(X,Y) = 0 which contains bxq points (x,y) on a Galois field F_q with q elements, q being a power of a prime number and b being the highest degree in Y of the equation f(X,Y) = 0.

13. Device according to Claim 11 , characterized in that said algebraic geometry code is constructed on a curve f(X,Y) = 0 which contains a number of points (x,y) strictly less than bxq on a Galois field F_q with q elements, q being a power of a prime number and b being the highest degree in Y of the equation f(X,Y) = 0.

14. Device according to Claim 11 or 12, characterized in that it comprises means for writing any code word v, to be calculated from an item of information, in the form of a b-tuplet v = [v₀, ... , V_b-i] of q-tuplets v^ in a finite

Galois field F_q with q elements, b being a strictly positive integer, q being a power of a prime number and £ being an integer between 0 and b-1 , v satisfying the equation H.v^τ = 0, where H designates the parity-check matrix of said algebraic geometry code and ^τ designates the transposition, and in that said device also comprises means for defining b words Uj of length q in F_q, i being an integer between 0 and b-1 , as follows:

where Ϋ_e designates a diagonal matrix of size qxq having the element y_£ (γ^u_1) in position (u,u), u being an integer between 1 and q-1 , where γ is a primitive element of the field F_q and where the elements y₀(x), ... , yw(x) are the solutions in Y of an equation f(x,Y) = 0 for any element x of the field F_q.

15. Device according to Claim 11 or 13, characterized in that it comprises means for writing any code word v, to be calculated from an item of information, in the form of a b-tuplet v = [v₀, ... , vn] of words v_e of length n^ in a finite Galois field F_q with q elements, £ being an integer between 0 and b-1 , the n^ values satisfying n^ > n^₊ι , b being a strictly positive integer, q being a power of a prime number and v satisfying the equation H.v^τ = 0, where H designates the parity-check matrix of said algebraic geometry code and ^τ designates the transposition, and in that said method also comprises:

- means for supplementing each word v_e with ng - n^ zeros, and

- means for defining b words Uj of length no in F_q, i being an integer between 0 and b-1 , as follows: u₀ = v₀ +... + v_b_₁, u₁ = v₀ ^<r₀ +... +v_b_₁αr_b._{1 l}

(3)

' b-1 = v₀r₀ b-1 + ...+v_b_₁r_t b-1 b-1 ^■ where <ϊ_t designates a diagonal matrix of size n₀χno where the first n^ diagonal elements contain the (^ +1)^th solutions y^(x) associated with the n^ values of x for which the equation f(x,Y) = 0 admits at least Z +Λ solutions in Y.

16. Device according to Claim 14, characterized in that it comprises means for rewriting the equation H.v^τ = 0 in the form of a set of b equations, as follows:

where Ht designates a matrix of size txq, t being a positive integer, having the element γ^(l"1)(i~1) in position (i,j) for any integer i between 1 and t and for any integer j between 1 and q-1 , having the element 1 in position (1 ,q) and having 0 in the other positions in the last column and where, for any integer m between 0 and b-1 , t(m) designates the number of rows in the matrix Ht(m).

17. Device according to Claim 15, characterized in that it comprises means for rewriting the equation H.v^τ = 0 in the form of a set of b equations, as follows:

where, for ≤ £ ≤ b , H^ designates a matrix of size t( )χ n^ , t(£) being a positive integer, having in position (i,j), for 1 < i < t(£) and 1 < j < n^ , the element γ -i)(^j- ) ₎ where is the (j+1)^th of the elements x of F_q, for which the equation

f(x,Y) = 0 admits at least £ +1 solutions in Y.

18. Device for the systematic coding of an information sequence comprising k symbols on a finite Galois field F_q with q elements, q being a power of a prime number and k being a positive integer, in the form of a code word of length n on F_q, n being an integer greater than k, characterized in that said code word belongs to a code having as a parity-check matrix a matrix of the form:

H (1)

where Ht designates a matrix of size txq, t is a positive integer, the element in position (i,j) of H is γ^{( l)(i"1)} for any integer i between 1 and t and for any integer j between 1 and q-1 , the element in position (1 ,q) of H is 1 , the element in position (i,q) of H is 0 for i>0, γ is a primitive element of F_q, the t(m) satisfy t(m) > t(m+1) for m = 0, ... , b-2 and Υ_t is, for any integer £ between 0 and b-1 , a diagonal matrix of size qxq having the element y_{i u} in position (u,u), these elements y_{i u} all being different for a fixed £ .

19. Device for the systematic coding of an information sequence comprising k symbols on a finite Galois field F_q with q elements, q being a power of a prime number and k being a positive integer, in the form of a code word of length n on F_q, n being an integer greater than k, characterized in that said code word belongs to a code having as a parity-check matrix a matrix of the form:

Uα(0) uσ(1) μσ(b-1) ^πt(0) ^πt(0) ^πt(0) μσfb-I

H ^πt(1) ^z 0 ^πt(1) 1 ^πt(1) ^z b-1 (13) μσ(0) y-b-1 μσ(1) ^yb-1 μσ -I b-l

_ⁿt(b-1) ^z 0 ⁿt(b-1) ^z 1 ^πt(b-1) b-1 _ where the σ(f) are subsets of F_q satisfying σ(f) 3 σ(f+1) for f = 0, ..., b-2, H^ comprises t rows and a number of columns equal to the number of elements in σ, the element in position (i,i) of H is X_j ^1"1 where X_j represents the j^th element of σ, 0° represents 1 and the elements of the matrices Υ_e , £ = 0, ... , b-1 , which are associated with the same Xj are all different.

20. Digital signal processing apparatus, characterized in that it comprises means adapted to implement a coding method according to any one of Claims 1 to 10.

21. Digital signal processing apparatus, characterized in that it comprises a coding device according to any one of Claims 11 to 19.

22. Telecommunications network, characterized in that it comprises means adapted to implement a coding method according to any one of Claims 1 to 10.

23. Telecommunications network, characterized in that it comprises a coding device according to any one of Claims 11 to 19.

24. Mobile station in a telecommunications network, characterized in that it comprises means adapted to implement a coding method according to any one of Claims 1 to 10.

25. Mobile station in a telecommunications network, characterized in that it comprises a coding device according to any one of Claims 11 to 19.

26. Base station in a telecommunications network, characterized in that it comprises means adapted to implement a coding method according to any one of Claims 1 to 10.

27. Base station in a telecommunications network, characterized in that it comprises a coding device according to any one of Claims 11 to 19.

28. Information storage medium which can be read by a computer or a microprocessor storing instructions of a computer program for implementing a coding method according to any one of Claims 1 to 10.

29. Information storage medium according to Claim 28, wherein said storage medium is partially or totally removable.

30. Computer program product comprising sequences of instructions for implementing a coding method according to any one of Claims 1 to 10.