FR2869482A1 - Decoding method for e.g. encoded digital signal reading apparatus, involves calculating associated word, whose corresponding aggregated words according to aggregates are associated aggregated words, in code of length n and dimension k - Google Patents

Abstract

The method involves calculating, from a received word, aggregated received words according to aggregate regrouping units of localization set. The aggregated word associated in Reed-Solomon code is obtained for each aggregated received word by list decoding method. The associated word, whose corresponding aggregated words according to the aggregates are associated aggregated words, is calculated in a code of length n and dimension k. Independent claims are also included for the following: (A) a data storage system having encoded digital signals recording apparatus, a storage medium and encoded digital signals reading apparatus comprising a decoding device (B) a data telecommunication system having digital signal transmission apparatus and encoded digital signal receiving apparatus (C) a fixed data storage unit comprising computer program code for executing a method of decoding (D) a partially or completely removable data storage unit comprising computer program code for executing a method of decoding (E) a computer program containing instruction for controlling a data processing device for implementing a decoding method.

Description

The present invention relates to communication systems or

  for recording data in which, in order to improve the fidelity of the transmission or storage, the data is subjected to channel coding. It relates more particularly to a decoding method, as well as

  devices and apparatus for carrying out this method.

  It is recalled that the channel coding consists, when forming the code words sent to a receiver or recorded on a data medium, to introduce some redundancy into the data. More precisely, the information initially contained in a predetermined number k of symbols taken from a finite-size alphabet q is transmitted by means of each code word; from these k information symbols are computed a number n> k of symbols belonging to this alphabet, which constitute the components of the codewords: v = (v1, v2, ..., vn). The set of code words obtained when each information symbol takes any value in the alphabet, constitutes a kind of dictionary called code of dimension k and length n.

  When the size q of the alphabet is a power of a prime number, we can give this alphabet a body structure, called Galois body, denoted Fq, whose non-zero elements can be conveniently identified as being equal to each other. to y 'for a corresponding value of i, where i = 1, ..., q 1, and where y is a root (q -1) th primitive of the unit in Fq.

  In particular, some codes, called linear codes, are such that any linear combination of codewords (with the coefficients taken from the alphabet) is still a codeword. These codes can, conveniently, be associated with a matrix H of dimension (nk) xn, called parity matrix: a word v of given length n is a code word if, and only if, it verifies the relation: H EVt = 0 (where the exponent T indicates the transposition); it is said that the code is orthogonal to this matrix H. At the receiver, the associated decoding method then judiciously uses this redundancy to detect possible transmission errors and if possible correct them. There is a transmission error if the difference e between a received word r and the corresponding code word v sent by the issuer is non-zero.

  More precisely, the decoding is done in two main stages.

  The first step is to associate the received word with an associated code word. Various methods have already been proposed to practically achieve this first step.

  According to a first possible method, the decoder first calculates the vector of error syndromes s = H E rT = H E eT of length (n - k), then determines an estimated value e of the error e in such a way that that (r) is a code mode. The vector (r - ê) then constitutes the associated code word.

  An alternative method for performing the first step is to search for all code words v that are located at a predetermined Hamming distance from the received word r. (Remember that the Hamming distance is by definition the number of components where two words of the same length have a different symbol.) This method is generally called list decoding (// st decoding in English), since we obtain in general a list of code words v that meet this definition.

  According to the different possible embodiments, it will then be possible to choose the code word associated with the received word r among the code words of the list by means of an additional criterion, or to consider all the codewords thus obtained in following the treatment.

  The second step is simply to invert the coding process, which makes it possible to obtain the sequence of k initial symbols (before coding) if the transmission errors have been correctly corrected.

  When using a list decoding method and no choice among the different code words of the list has been made beforehand, we reverse the coding process for all code words in the list and we get and a plurality of sequences of k possible symbols, from which one can choose the one that satisfies a predetermined criterion.

  Note that in the context of the present invention, we will often talk, for short, decoding to designate only the first of these steps, it being understood that the skilled person is able without difficulty to implement the second step.

  An important parameter in the characterization of a code is its minimum distance d, which can be defined as the smallest Hamming distance between two different words of this code. More precisely, it is in principle possible to find the position of possible errors in a received word, and to provide the correct replacement symbol (ie, identical to that sent by the sender) for each of these positions, whenever the number of erroneous positions is at most equal to INT [(d -1) / 2] (where INT denotes the integer part) for a minimum distance code d (for certain error configurations, it is possible to even sometimes do better). In any case, however, it is only a matter of principle, since it is often difficult to develop a decoding algorithm that achieves this performance. It will also be noted that, when the chosen algorithm manages to propose a correction for the received word, this correction is all the more reliable (at least, for most transmission channels) that it concerns a smaller number of positions. .

  Known codes include the so-called Reed-Solomon codes, which are known for their effectiveness. These are linear codes whose minimum distance d is equal to (n-k + l) when they are not extended. The parity matrix H of the Reed-Solomon code of dimension k and of length n (where n is necessarily equal to (q-1) or to a divisor of (q -1)) is a matrix at (n - k) lines and with n columns, which has the structure of a Vandermonde matrix. This parity matrix H can be defined for example by taking H1 = a'1-1) (1 <_ i <n - k, 1 j s n), where a is a root n of the unit in Fq; we can then label the component vj, where 1 <j n, of any code word v = (v1, v2, ..., vn) by means of the element a '' - 1) of Fq; this is why a set such that (1, a, a2, ..., an-1) is called the locating set in English of the Reed-Solomon code.

  It is furthermore known that such a Reed-Solomon code can be equivalently defined as follows: the set of words of the Reed Solomon code is the set of words (p (1), p (a), p (a2) ..., p (a - ')) when p traverses the set of polynomials of degree at most (k -1) on Fq.

  For the decoding of Reed-Solomon codes, the Berlekamp-Massey algorithm for error localization and the Forney algorithm for error correction are commonly used.

  An alternative to these methods, which applies list decoding to Reed-Solomon codes, was proposed by Dr. Sudan in his article Decoding of Reed Solomon Codes beyond the Error-Correction Bound, (Academic Press, 1997).

  According to this article, the problem of the decoding by list of a received word r = (r1, r2, ..., r) returns to the search for the set of polynomials f of degree at most (k -1) on Fq which generate a code word (f (1), f (a), f (a2) ..., f (a ^)) situated in a radius δ of the received word r in the sense of the Hamming distance.

  The problem of decoding by list therefore reduces to the following problem: find the set of polynomials f of degree at most (k -1) on Fq, for which the number of components i such that f (a'- ') = r, is at least equal to (n-5). It is therefore an interpolation problem since we are looking for polynomials f which satisfy f (x) = y for at least (n-8) pairs (x, y). To solve this problem of interpolation in a practical way, the article by M. Sudan proposes to look for a polynomial Q (X, Y) (where X and Y are elements of Fq) which vanishes in every point (a ' - ', r;), for i between 1 and n, then to factorize the polynomial Q (X, Y) into irreducible factors, in order to find factors of the form (y - f (x)). The functions f thus obtained are candidates for the solution of the interpolation problem since they satisfy r; - f (a'- ') = 0 for some values of i at least.

  Naturally, knowledge of at least one polynomial solution of the interpolation problem makes it easy to obtain the corresponding code word (f (1), f (a), f (a2). ')), which can be used as a codeword associated with the received word r.

  In modern information media, such as computer hard drives, CDs (compact discs) or DVDs (digital video discs), there is an attempt to increase the density of information. When such media is affected by a physical defect such as a scratch, a large number of information symbols may be rendered illegible. However, this problem can be remedied by using a very long code. However, as indicated above, the length n of the words in the Reed-Solomon codes is less than the size q of the symbol alphabet. Therefore, if one wishes to have a Reed-Solomon code having long codewords, large values of q should be considered, which leads to expensive calculations and of memorization. In addition, large values of q are sometimes unsuitable for the intended technical application. This is why we have tried to build codes that naturally offer a greater length of words than Reed-Solomon codes without requiring a larger alphabet.

  In particular, we have recently proposed codes called algebraic geometry codes or geometric Goppa codes (see for example the article by Tom Hoholdt and Ruud Pellikaan entitled On the Decoding of Algebraic-Geometric Codes, IEEE Trans. No. 6, pages 1589 to 1614, November 1995). These codes are constructed from a set of n pairs (x, y) of symbols belonging to a selected Galois field Fq; this set of couples constitutes the set of location of the algebraic geometry code. In general, there exists an algebraic equation with two unknowns X and Y such that the pairs (x, y) of this set of localization are all solutions of this algebraic equation. The values of x and y of these pairs can be considered as the coordinates of points Pfl (where /3 = 1, ..., n) forming an algebraic curve.

  An important parameter of such a curve is its genus g.

  In the particular case where the curve is a straight line (the genus g is then zero), the algebraic geometry code is reduced to a Reed-Solomon code. For q and g given, some algebraic curves, called maximum, allow to reach a length equal to (q + 2g J), which can be very high; for example, with an alphabet size of 256 and a gender of 120, code words of length 4096 are obtained.

  In the context of the present invention, we are interested in a very general class of codes containing the codes of algebraic geometry: these codes, an example of which is described in detail below, are defined by a parity matrix of the following way. At any monomer Y'X, where i and j are positive or null integers, we associate a weight. If, for an integer p 0, there exists at least one monomial whose weight is p, we say that p is a realizable weight. Let pi <P2 <eE <Pn_k be the (nk) smaller weights that are realizable, and let ha (where a = 1, ..., nk) be a monome of weight pa, the monomials ha (for a = 1 ,. .., nk) all having a distinct weight. The inline element a and column fi of the parity matrix is equal to the monomial ha evaluated at a point PP (where fi = 1, ..., n) of a set of pairs of FgxFq, which therefore constitutes the set of code localization. Each point Pjg then serves to identify the fi th component of any code word.

  Like all codes, algebraic geometry codes can be modified and / or shortened. We say that a given code Cmod is a modified version of the code C if there exists a non-singular square diagonal matrix A such that each word of Cmod is equal to v É A with v in C. It is said that a code given is a shortened version of the code C if it understands only the words of C of which, for a number R of predetermined positions, the components are all null: these positions being known of the receiver, one can dispense with transmitting them, of so that the shortened code is of length (n R). In particular, it is common to shorten an algebraic geometry code by removing from the location set, if necessary, a point, or several points, whose x coordinate is zero.

  The algebraic geometry codes are advantageous as to their minimum distance d, which is at least equal to (n - k + 1 - g), and, as said, as to the length of the codewords, but they have the disadvantage of requiring quite complex decoding algorithms, and therefore quite expensive in terms of equipment (software and / or hardware) and processing time. This complexity is in fact greater or lesser depending on the algorithm considered, a greater complexity being in principle the price to pay to increase the decoder error correction capacity (see for example the article by T. Hoholdt and R Pellikaan quoted above). In general, the higher the algebraic curve used, the longer the codewords are, and the more complex the decoding.

  As for the Reed-Solomon codes, methods for decoding a received word r = (r1, r2, ..., rn) have already been proposed that use an error localization algorithm and then a correction algorithm. errors.

  This type of method is described, for example, in the article by AN Skorobogatov and SG Vladut entitled On the Decoding of Algebraic Geometric Codes (IEEE Trans., Theory, Vol 36 No. 5, pages 1051-1060, November 1990) for localization. and A Generalized Formula for Algebraic Geometric Codes, Douglas A. Leonard (IEEE Trans., Theory, 42, 4, pages 1263-1268, July 1996) for error correction. .

  These algorithms are generally complex to implement, in particular because they include the multiplication of two-variable polynomials, in addition to formal multiplications in Fq.

  In order to try to benefit from the advantages proper to list decoding, it has been proposed to apply the method of list decoding to algebraic geometry codes. The principles of this decoding method are described in V. Guruswami's Improved Decoding of Reed-Solomon and AlgebraicGeometry Codes, and M. Sudan (IEEE Trans., Theory, Vol 45, No. 6, September 1999).

  A list decoding method which incorporates these principles and specifies the interpolation techniques used is also described in the article Decoding Hermitian codes with Sudan's algorithm by T. Hoholdt and R. Refslund Nielsen (Applied Algebra, Algebraic Algorithm and Error Correcting Codes reading notes in computer science, Springer Verlag 1999).

  By analogy with the interpolation method used for Reed-Solomon codes 5, the techniques proposed in these articles include the search for a polynomial Q (X, Y, Z), where X, Y and Z are elements of Fq, which vanishes in every point (PQ, re) for, 6 = 1, ..., n, then the factorization of the polynomial Q (X, Y, Z). (Recall that the r, are the components of the received word r and that the points P) g of Fq2 form the set of localization of the code.) These techniques prove however cumbersome to implement, in particular because the polynomial Q (X, Y, Z) to factorize is a polynomial with three variables taking their values in a Galois body. The list decoding methods therefore did not seem to provide a solution to the problem of decoding the algebraic geometry codes that is practically realistic.

  Under these conditions, the invention proposes a decoding method for code of length n and of dimension k on a Galois field Fq, the code being defined by means of a set of localization of cardinal n on Fq x Fq, characterized in that, the location set comprising at least p subsets called aggregates, where, u 2, each aggregate grouping the elements (x, yp (x)) of the location set having a common value of x, for p = 0, (x) -1, where ,, (x) is the cardinal of the aggregate, it comprises the following steps: - calculation, from a received word r, of aggregated received words rf according to said p aggregates; for each received received word ri, obtaining at least one associated aggregated word r; in a Reed-Solomon code by a list decoding method; calculating at least one associated word in the code of length n and of dimension k, whose corresponding aggregated words according to said aggregates are the associated aggregated words Ej In order to solve, in particular, the problems mentioned above, the invention proposes to use a code whose set of localization can be divided into subsets called aggregates. By definition, an aggregate groups the pairs (x, y) of the localization set with a common value of x (we could have defined the aggregates as well by a common value of y by exchanging the roles of the unknowns X and Y of the equation representing the algebraic curve on which the code is defined). When we want to make this structure appear in aggregates, the pairs of the localization set (which is not necessarily a set of solutions of an algebraic equation of type C (a, b)) will be noted. (x, yp (x)), where p = 0, ..., 2 (x) -1 and 2 (x) is the cardinal of the considered aggregate, and the components of any word c of length n will be denoted c (x, yp (x)); it will be said that the components of c which, labeled in this way, have the same value of x form an aggregate of components of the word c. The invention thus proposes a method and a decoding device whose complexity is reduced. By complexity, here means the number of operations required to perform the decoding, or, in the case of an embodiment in a decoding circuit, the number of logic gates necessary for its implementation.

  According to a practical solution to be implemented, the code is defined by a parity matrix H whose elements Haa for 1 <, 5 n are equal to the value taken by a monomer ha = Y'X 'on the points of the set of localization, the monomials ha for 1 a <nk all having a distinct weight.

  The problem of decoding such a code is reduced to a plurality of decodings for Reed-Solomon codes. It is thus possible to benefit from the advantages offered by the list decoding technique, the implementation of which for Reed-Solomon codes is much less complex than for the algebraic geometry codes.

  It will also be noted that the system of equations, which makes it possible to obtain the codeword associated with the word received from the associated aggregated words, is a Vandermonde system; as it is well known to those skilled in the art, the resolution of such a system of linear equations is particularly simple.

  A special case of this solution is the case of an algebraic geometry code.

  Thus, thanks to the invention and contrary to what one could think, one can benefit from the advantages offered by the method of decoding by list, without however suffering from the heaviness implied by the use of polynomials with three variables taking values in a Galois body.

  The list decoding method generally comprises, for each received aggregate word r1, the following steps: searching for at least one polynomial f1 over Fq of degree at most k1 such that the number of components r1 (x) of the received word aggregated rj for which r1 (x) = f1 (x) is at least equal to a number r / l, where k1 is a predetermined integer and where rit, u and; calculating the associated aggregated word r such that i-1 (x;) = f1 (x,) for all i between 1 and u.

  The number rit may be predetermined or correspond to the maximum number of components for which r1 (x) = g1 (x) when g1 traverses the set of polynomials over Fq of degree at most k1.

  Finally, when the step of searching for at least one polynomial fi generates at least two solution polynomials, a possible implementation of the invention consists in carrying out the following steps: for each solution polynomial, calculating an aggregate word partner; for each solution polynomial, calculating at least one associated word as a function of said associated aggregated word calculated for the solution polynomial; 2869482 11 - comparison of the associated words calculated for each solution polynomial with the received word r - selection of the associated word Y 'among the associated words calculated for each solution polynomial.

  The invention proposes in the same order of idea a decoding method for code of length n and of dimension k on a Galois field Fq, the code being defined by means of a set of localization of cardinal n on Fq x Fq , characterized in that, the location set comprising at least, u subsets called aggregates, where, u 2, each aggregate grouping the elements (x, yp (x)) of the location set having a common value of x, for p = 0, (x) -1, where .1, (x) is the cardinal of the aggregate, it comprises the following steps: - calculation, from a received word r, of received words aggregates = [ri (xj), ri (x2), ..., ri (xp) j for j = 0, ..., Jmax 'where J, nax 1, according to a formula of the type ri (x) = A (x) [r (x, yo (x)), ..., r (x, y, I (X, l (x))], where A (x) is a function on FgA (X), for each aggregate received word ri, search for at least one polynomial fi over Fq of degree at most ki such that the number of components ri (x) of the aggregate received word ri for which ri (x ) = fi (x) is at least equal to a number i, where iu - for each aggregate received word ri, calculate at least one associated aggregate word such that fi (x;) for any i between 1 and, u - calculating at least one associated code word ï i whose components i '(x, y) are such that A; (x,) [N (x; yo (x y2 (x,) _ 1 (x;))] = i ^ i (x;) for all i between 1 and, u and for all j between 0 and jmax In the embodiment already mentioned above and proposed hereinafter, in which the parity matrix is formed of the values taken by the monomials Y'X 'on the localization set, advantageously brings together the monomials having the same power of the variable Y in sets M ;: Mi = {Y'X1 IO <- i 5 (pj -bj) / a} for 0 <j jmax, or Imax <a. The set Mi thus has for cardinal: t ( j) = 1 + INT [(pi -bj) / aJ.

  Let x1, x2, ..., xp denote the different values of x in the localization set, and by v = [v (xl, Yo (x1)), ..., v (x1, .ya11 1 ( x1)), ..., v (x, ya, 1 (xv)) 1, any code word. We can then construct, for each aggregate attached to one of the values x1, x2, ..., xp of x, (jmax + 1) aggregated symbols .1. (X) -1 vj (x) {yp (x )] iv (x, yp (x)) P = 0 for j = 0, ..., jmax. These aggregated symbols are used to form (jm +1) aggregated words v j = [v j (xt), v j (x2), ..., vi (xp) l of length g.

  It is easy to verify that the condition of belonging to the algebraic geometry code (ie H EVt = o) is equivalent to the set of (jmax +1) equations: Ht (.i). vT = o, where the function t (j) is given above and, by definition, 1 1 1 x1 X2 t-1 t-1 t-1 x1 x2 x1i - The interest of this formulation is that this matrix reads of the equation is a matrix of Vandermonde defined on Fq; therefore, if we consider H j) as a parity matrix defining codewords vi, it is, for each value of j, a Reed-Solomon code, for which decoding algorithms are also known. simple than performant.

  In the proposed embodiment, for j = 0, ..., jmax, the aggregated received words rl = [ri (xI), ri (x2), ..., ri (xii)] are therefore expressed as functions of the received word r by the formula ri (x) _E [yp (x)] 'r (x, yp (x)) (1) p = o The implementation of this decoding method is particularly advantageous for a certain type of channels: these are the channels in which the data to be transmitted are grouped into blocks of predetermined length, and in which the error rate per transmitted data is essentially constant within the same block. In other words, such channels are physically characterized in that, most often, transmission noises affect block data, and may differently affect different blocks; thus, for some blocks, the probability of error may be very low or even zero, while for some other blocks the probability of error may be high and even close to (q -1) / q. This translates into a flurry of errors (burst of errors in English) in terms of Galois body symbols.

  An important example in industrial practice of such channels is hard disk recording / playback.

  Indeed, the bits constituting the symbols of the code words are usually recorded by means of a modulation code intended to ensure certain desirable spectral properties, for example the property according to which the number of 1 is, on average, approximately equal to the number from 0. To obtain this result, the bits entering the modulator are grouped in blocks of N bits; depending on the balance between the total number of 1 and the total number of 0 previously recorded, a block a will be recorded on the disk, either as such, or in the form of its complement a '(where we changed each 1 in a 0 and vice versa), so that the new balance sheet is as close as possible to equality.

  When a recording error occurs on a particular bit, the ambiguity resulting from the modulation coding means that this bit can not be individually corrected, and therefore, in practice, it must be resolved to consider the entire block of N bits as erroneous. Since it is also considered that the bits recorded and read form bytes representing symbols of a Galois field, an erroneous block of N bits will generally straddle several symbols; for example, if N = 48 and q = 210, an erroneous block can be straddling 6 bytes of 10 successive bits, which results in a burst of errors affecting 6 symbols of the Galois body.

  It can therefore be seen that the proposed decoding method which first corrects erroneous aggregates associated with the received word and not erroneous individual components of this word, is well adapted to take advantage of such a distribution of noise on a transmission channel: it suffices to insert in the adjacent position in the data stream to be transmitted the components of a code word (channel) belonging to the same aggregate.

  On the other hand, and for the same reason, the number of individual errors that can be corrected with this method may be less than the theoretical error correction capability of the code (as explained above, this theoretical capacity is equal to INT [(d -1) / 2], where d is the distanceminimal of the algebraic geometry code considered). However, the question arises as to whether the codes that can benefit from this decoding process all suffer from this disadvantage to the same extent.

  Naturally, the invention also relates to decoding devices embodying the invention.

  The advantages of these decoding devices are essentially the same as those of the correlative methods described above.

  The invention also relates to: an encoded digital signal reading apparatus comprising a modulated data reader, means for demodulating said modulated data into coded digital signals, and a decoding device as described above; receiving coded digital signals comprising a modulated data receiver, means for demodulating said coded digital signal modulated data and a decoding device as described above, - a data storage system comprising at least one data storage apparatus recording coded digital signals, at least one recording medium and at least one coded digital signal reading apparatus as described above, - a data communication system in block form of predetermined length, comprising at least one digital signal transmitting apparatus and at least one digital signal receiving apparatus s as described above, - immovable data storage means comprising computer program code instructions for performing the steps of any of the decoding and / or communication methods discussed above; partially or fully removable data storage means having computer program code instructions for performing the steps of any of the decoding and / or communication methods discussed above, and - a program of computer, containing instructions such that, when said program controls a programmable data processing device, said instructions cause said data processing device to implement one of the decoding and / or communication methods set forth above.

  The advantages offered by these reading or receiving devices, these data storage or telecommunication systems, these data storage means and this computer program are essentially the same as those offered by the decoding methods according to the invention. .

  Other aspects and advantages of the invention will appear on reading the detailed description below of particular embodiments, given by way of non-limiting examples. The description refers to the accompanying drawings, in which: - Figure 1 is a block diagram of an information transmission system in the context of the invention, - Figure 2 shows a recording apparatus of digital signals incorporating an exemplary encoder, - Figure 3 shows a digital signal reading apparatus incorporating a decoding device according to the invention, - Figure 4 shows the general diagram of the decoding technique proposed by the invention, - FIG. 5 illustrates an embodiment of the interpolation phase proposed by the invention; FIG. 6 illustrates an embodiment of the factorization phase proposed by the invention; FIG. 7 describes a possible solution when the previous phases give several solutions.

  FIG. 1 is a block diagram of an information transmission system using channel coding according to the invention.

  This system has the function of transmitting information of any kind from a source 100 to a recipient or user 109. First, the source 100 puts this information in the form of symbols belonging to a certain alphabet (for example bits bytes), and transmits these symbols to a storage unit 101, which accumulates the symbols so as to form sets each containing k symbols.

  Then, each of these sets is transmitted by the storage unit 101 to an encoder 102 which incorporates redundancy, so as to construct a word of length n belonging to the chosen code.

  The code words thus formed are then transmitted to a modulator 103, which associates with each symbol of the code word a modulation symbol (for example, a series of bits according to a modulation code, as explained above). Then, these modulation symbols are transmitted to a recorder (or transmitter) 104, which inserts the symbols into a transmission channel. This channel can be for example a storage on a suitable medium such as a DVD or a magnetic disk or a magnetic tape. It can also correspond to a wire or non-wire transmission as is the case of a radio link.

  The transmitted message reaches a reader (or receiver) 105, after having been affected by a transmission noise whose effect is to modify or erase some of the modulation symbols.

  The reader (or receiver) 105 then transmits these symbols to the demodulator 106, which transforms them into symbols of the alphabet mentioned above. The n symbols resulting from the transmission of the same code word are then grouped into a word received in an error correction unit 107, which implements a decoding method according to the invention, so as to provide a word associated code. Then this associated code word is transmitted to a redundancy suppression unit 108, which extracts k information symbols by implementing a decoding algorithm inverse to that implemented by the encoder 102. Finally, these symbols of information is provided to the recipient 109.

  Units 107 and 108 may be considered together to form a decoder 10.

  The block diagram of FIG. 2 very schematically shows an information data recording apparatus 48 incorporating the encoder 102.

  This apparatus 48 comprises a keyboard 911, a screen 909, an external information source 100, a modulator 103 and a modulated data logger 104, jointly connected to input / output ports 903 of a coding device 102 which is realized here in the form of a logical unit.

  The coding device 102 comprises, interconnected by an address and data bus 902: a central processing unit 900, a random access memory RAM 904, a read only memory 905, and said input / output ports 903.

  Each of the elements illustrated in Figure 2 is well known to those skilled in the field of microcomputers and transmission systems and, more generally, information processing systems. These known elements are therefore not described here. It is observed, however, that: the information source 100 could be, for example, an interface device, a sensor, a demodulator, an external memory or another information processing system (not shown), and could for example provide signal sequences representative of speech, service messages or multimedia data including IP or ATM type, in the form of binary data sequences, and - the recorder 104 is adapted to record data modulated on a support such as a magnetic disk.

  The random access memory 904 stores data, variables and intermediate processing results, in memory registers carrying, in the description, the same names as the data whose values they retain. It will be observed, in passing, that the word register designates, through the present description, both a low capacitance memory area (a few binary data) and a large memory area (for storing an entire program) within 'a RAM or a ROM.

  The random access memory 904 comprises in particular the following registers: an information symbol register in which the information symbols belonging to Fq are stored, and a code word register, in which the code words v are stored before they are subject to modulator 103.

  The read-only memory 905 is adapted to keep, in registers 25 which, for convenience, have the same names as the data they retain: the operating program of the central processing unit 900, in a program register; the length of the code words, in a register n, - the cardinal of the Galois field Fq serving as an alphabet for the code used, in a register q, - the number of information symbols used to construct a codeword, in a register k, and - the parity matrix of the code, in a register H.

  Naturally, the description which has just been made represents only an illustrative and nonlimiting example of embodiment of an encoder.

  The block diagram of FIG. 3 represents a coded digital signal reading device 70 incorporating the decoder 10. This apparatus 70 comprises a keyboard 711, a screen 709, an external information recipient 109, a data reader 105 and a demodulator 106, jointly connected to input / output ports 703 of the decoder 10 which is embodied herein as a logical unit.

  The decoder 10 comprises, interconnected by an address and data bus 702: a central processing unit 700, a random access memory (RAM) 704, a read only memory (ROM) 705, and said ports input / output 703.

  It is observed that: the information recipient 109 could be, for example, an interface device, a display, a modulator, an external memory or another information processing system (not shown), and could be adapted to receive sequences of signals representative of speech, service messages or multimedia data, particularly of the IP or ATM type, in the form of binary data sequences, the reader 105 is adapted to read data recorded on a medium such that a magnetic or magneto-optical disk.

  The RAM 704 stores data, variables and intermediate processing results, in memory registers carrying, in the description, the same names as the data whose values they retain. The random access memory 704 comprises in particular the following registers: received word registers, in which the received words are stored, - aggregate word registers, in which the aggregate received words are stored, - an estimated symbol register, in which the symbols are kept derived from a word received during correction, - an associated word register, in which the symbols of the associated code words are stored, and - an information symbol register, in which are kept the symbols resulting from the suppression of the redundancy.

  The read-only memory 705 is adapted to keep, in registers which, for convenience, have the same names as the data they retain: the operating program of the central processing unit 700, in a program register, the length code words in a register n, - the cardinal of the Galois body Fq serving as an alphabet for the code used, in a register q, and - the number of information symbols used to construct a code word, in a register k.

  Note that in embodiments where only a list decoding method is used, there is no need to memorize the parity matrices of the Reed-Solomon codes. Indeed, the method of decoding by list uses the values of the set of location and not the parity matrix of the code, as we will see in the following.

  The storage of the parity matrix may, however, be useful, for example if it is desired to use a method combining the list decoding proposed by the invention and a decoding of another type.

  It should be noted that in some applications it will be convenient to use the same computing device (operating in multi-tasking mode) for the exchange, i.e. both transmission and reception, of signals according to the invention; in this case, the units 10 and 102 will be physically identical.

  An application of the invention to the mass storage of data has been described above by way of example, but it is clear that the methods according to the invention can equally well be implemented within a network of data. telecommunications, in which case the unit 105 for example could be a receiver adapted to implement a packet data transmission protocol on a radio channel.

  An exemplary embodiment of the invention will now be described by way of illustration in the case of a particular algebraic geometry code.

  We consider here an algebraic geometry code on F16 based on the algebraic curve C (2.5) in F16xF16 represented by the equation f (X, Y) = 0 with f (X, Y) = Y2 + Y + X5 points of the curve C (2,5) (that is to say the pairs (x, y) solutions of the equation f (X, Y) = 0) thus constitute the set of localization of the code.

  In the present case, the equation Y2 + Y + X5 = 0 has thirty non-trivial solutions in F16xF16 which are the following: (1, a), (1, a10), (a, a), (a, a4) , (a2, a), (a2, a), (a3, a5), (a3, a1), (a4, a1), (a4, (y4), (a5, a), (a5, ( 2), (a6, a), (a6, a10), (a7, a), (a7, a4), (a8, (12), (12, a, '), (a9, a5), (a9 , a1), (a1o, a), (a10 a4) (a11a) (a11a) (a12a), (a12, a1), (a13, a), (a13a4) (a14, a2), a14, a8), where a is the generating element of F16 such that a4 + a + 1 = 0.

  The location set can here be divided into, u = 15 aggregates each grouping the points (x, y) having a common value of x. In the case presented here, the cardinal of each aggregate is equal to 2.

  Taking a more general formulation, if xi, ..., x are the values of x associated respectively with each aggregate (here x, = a'd), the ith aggregate consists of 2 (x,) elements (here / 1, (x1) = 2): (x,, yo (x,)) ,.

  , (x,, y2 (X,) _1 (x,)) ... DTD: We can therefore define an algebraic geometry code of length n = 30 and dimension k = 20 by a parity matrix H of dimensions n = 30, n-k = 10 whose elements are defined as follows: 25), J (I, P) = for i = 1, u, p = 0, ..., /1.(x,) -1 , j = 0, ..., Jmax and z = 0, ..., t (j) -1 i-1 where 1 (-z-, j) = z + Et (l) for j> 1 and I ( z, 0) =, f = o and J (i, p) = i +, u. p. The function t used is the one defined above: it defines the number of monomials of the form Y'XT to j fixed. On the other hand, it has been assumed to simplify the notations that all the aggregates have the same cardinal / t (x,) = as is the case in the example studied here.

  The parity matrix H is thus formed by the evaluation of a monomer Y'XT associated with each line on each point of the location set, each point being associated with a column.

  In the parity matrix H obtained precisely according to the formula above, the lines associated with the monomials Y'XT of the same power j are advantageously grouped and classified according to the order of the powers z. Similarly, the columns are grouped into a subset of columns, so that each subset contains one and only one column associated with each aggregate; moreover, in each subset of columns, the columns are classified in the order of the aggregates (x ,, ..., xp).

  By using matrices H of the form 1 1 1 Ht XI X2 .. Xp, the parity matrix H can thus be written t-1 t-1 t-1 x1 x2 x -

_

  ## EQU1 ## where each Yp is a diagonal matrix of diagonal elements (.vp (x1), ..., yp (x,)).

  In the example studied here, the parity matrix H can thus be written H6 H6 with the following matrix for H6 H4Y0 H4Y H = H1 (') Yo H = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 a a2 a3 a4 a6 a7 a9 a aa a2 a3 a4 a2 a4 a6 a8 a2 a4 a3 a7 a9 a1 a13 1 a3 a6 a9 a12 1 a3 a6 a9 a12 1 a3 a6 a9 a12 1 a4 a8 a12 a as a9 a'3 a2 ab a 'a14 a3 a7 al] 1 a5 a10 1 a5 a10 1 a a5 a5 as 1 a5 a5 1 1 H4 following matrix 1 1 1 1 1 1 1 1 1 1 1 1 1 1 a a2 a3 a4 a5 a6 a7 a8 a9 a 'a "a'2 a'3 a'4 1 a2 a4 a6 a8 a' a'2 a14 a a3 a5 a7 a9 to a3 3 a3 a6 a9 a12 1 a3 a6 a9 a12 1 a3 a6 a9 a12 and the respective diagonals of y and y being (a5, a, a2, a5, a, a2, a5, a a2 5, a, a 2 a 5 to a 2) and (a 1 a 4 a 8 a 10 a 4 a 10 a 4 a 10 a 4 a 8 a 10 a 4 a a).

  The different decoding steps as proposed by the invention will now be described, when it is desired to decode a received word r, and consequently to find at least one word of the algebraic geometry code associated with the received word r.

  This description will be made in particular with reference to Figure 4 which schematically describes the different decoding steps. The diagram of FIG. 4 may also represent an architecture that can be envisaged for an electronic device capable of implementing the invention. The invention is however not limited to devices of this type.

  The received word r therefore corresponds to a code word c after transmission in the channel, and thus possibly incorporates transmission errors.

  By way of example, it can be assumed that the code word c = (a9 a12 a13 a5 a12, a9, 0, 0, 0, 0, 0, 0, 0, a7, a6, a14, a2, 0, 0, 0, 0, 0, 0, 0, 0, 0) has been transmitted in the channel and the corresponding received word r has three errors indicated in bold below: r = (a9 a12 a13, a8, a12, a9, aO, 0, 0, 0, 0, 0, 0, 0, a7, (16, a14, a 0,0,0,0,0, a12,0,0,0).

  As we have seen, the decoding of the received word r consists mainly in finding the code word (s) associated with the word closest to the received word r in the sense of the Hamming distance.

  Given the code used, and in particular the particular form of the parity matrix, we note, however, that the membership condition of the codeword associated with the code (H = FT = 0) can be written in the form Ht (o ) HI (0) Hrro) Ht (1) .Y Ht (1) Ht (1) .Y-1 (r (xl, y0 (x1)), ..., i '- (xp, .y0 (xr) )) T = 0, Hr (imax) yimax Ht (imax) yimax 0 1 ... Ht (imax) yimax / 1 1 (r- (x,, ya, -1 (x1)), ..., r (xM, Y 1 (xp)) T i.e. - t 1 Ht (0) .E (r (x1, y (xl)), ..., r (xi,, .yp (xa) )) TP = 0 2 1

T 2 1

  j.% t (imax) ypmax, Yp)) (x1, ...,? (x, yp (x)) T tp = 0 As we have seen, defining for j = 0, ... , jmax aggregated words A 1 associated ri = [i (x1), i'i (x2), ..., ri (xi,)] = E Yp. (r (xl, yp (xl)), .. ., (xp, yp (xp)) T the p = 0 condition of belonging to the algebraic geometry code is reduced to 15 j, n +1 equations of the form Ht (i) & = 0, which each mean that the associated aggregated word ri is a word of the Reed-Solomon code having the parity matrix Ht (i), thus reducing the decoding of the received word r to the decoding of furax +1 aggregated words ri defined by the formula 2 1 ri (x ) = E [Yp (x)] ir (x, yp (x)) p = 0H .E yp E (r (x1, yp (x1)), ..., (xp, Y p (x,) ) p = 0 = 0 The decoding of the received word r can be performed separately for each aggregated word, possibly in parallel, as represented symbolically by the presence of several (precisely jmax +1) processing zones 310, ..., 320 in FIG.

  The first step of the processing relative to each aggregated word rj naturally consists in calculating the aggregated word (ie its components), which amounts in practice to sums of symbols of the received word r with different weights, such as indicated on blocks 315a, ..., 315b. Each block 315a,..., 315b thus generates an aggregated word ri.

  In the case presented here by way of example, the aggregated received words are therefore: ## EQU1 ## (a ', a12, a, a, a12, a9, O, O, O, O, O, O, O, O) + (a, 7, a, 6, a, 14, a 0,0,0,0,0, a12, 0,0,0) = (a, a4, a2, a ", a12, a9, a, 0, 0.0, a12, 0.0, 0.0) and r2 = Yp (a9, a12 al3 a8 al2 a9 0, o, oo, o, o 0) + y (a7 a6 a14, a , 0,0,0,0,0, a12, 0, o, 0) = (a13, a9, a9, 0 (.4 al3, a11, a5, 0, 0, 0, a, 0, 0, , 0) Within each zone 310,..., 320, the processing of each aggregated word ri is continued by its decoding according to a list decoding method, which comprises an interpolation phase and a factorization phase.

  The interpolation phase, which will now be described in detail with reference to FIG. 5, consists in finding a polynomial Q on FqxFq (here on F16xF16) which passes through the points (x ,, r] (x;)) , for i = 1, u, with an order of multiplicity m.

  We recall that a polynomial P (X, Y) passes through a point (x, y) with an order of multiplicity m if all its derivatives of Hasse of order (m-1) cancel each other at this point. If the a. are the coefficients of each monomial X'Y '00 0o i j'! within the polynomial P, this condition is expressed E E-k x y -u, k '1 = k' J = l 'for all k' and the positive integers such that k '+ l' <m.

  In the particular case where m = 1 taken for illustrative purposes, the interpolation is reduced to the search for a polynomial Q (X, Y) which passes through the points (x ,, rj (x;)), for i = 1, u, that is to say, which vanishes in these points.

  The interpolation algorithm used in the invention is based on the algorithms proposed by Arshad Amesh's VLSI Architecture for softdecision decoding of Reed-Solomon Codes, Ralf Koetter and Naresh R. Shanbag, Preprint, in IEEE transaction on VLSI systems. , February 2003, and Decoding Hermitian codes with Sudan's algorithm, T. HoHoldt and R. Refsund Nielsen, in Applied Algebra, Algebraic Algorithm and Error Correcting Codes, Reading notes in computer science, Springer Verlag 1999 In the description of the algorithm that follows , we use the parameter T defined by the formula T = Int (1 + 1, \ il + 8 * C) -1 where C = pm2 and Int the function 2 nk 1 integer part.

  As we shall see, this parameter T represents the number of candidate polynomials obtained by the interpolation algorithm. We will also use an index L which will make it possible to designate one of the polynomials among these T candidate polynomials.

  The interpolation algorithm starts at step E510 where the index L (L = 0) is initialized.

  We then go to step E511 where it is checked whether the index L is less than or equal to the number of candidate polynomials T. If the index L is smaller than the number of candidates T, we proceed to step E501 where the polynomial ZL is defined by ZL = YL.

  We then go to step E512 where the value of L. is incremented. Once the value L is incremented, we return to the step E511 in which we compare the index L to the total number T of candidate polynomials.

  When in step E511, we no longer check that L is less than or equal to T, we go to step E513 to which the value of a parameter i is initialized to zero.

  It may be noted that the steps E510, E511, E501 and E512, which define a loop on the value of the index L, make it possible to construct a base of polynomials ZL which is initialized by the base of polynomials YL.

  At step E513, the index i is initialized, which designates the different points where it is desired for the polynomial Q (X, Y) to vanish.

  After step E513, the index i therefore denotes a particular point.

  It is verified in step E514 that the last point has not been reached, by comparing the index i with the number p.

  If the index i has reached a value greater than, that is to say that all the points have been previously treated, proceed to the step E505 which will be described later.

  If, on the other hand, the index i is less than or equal to, the current point designated by this index i is processed by first going to step E515.

  In step E515, the value A is initialized to zero, which corresponds subsequently to the power according to the variable X of the monomium in question.

  We then go to step E516 where the current value A is compared with the multiplicity m (i) for the index point i considered. If the power A of the variable X in the considered monomial is strictly less than the multiplicity m (i), the process is continued for point i at step E518. If, on the other hand, the exponent A is equal to the multiplicity m (i), we go to the step E517 where the value of i is incremented, which amounts to passing to the processing of the next point.

  In step E518, the variable B is initialized to zero, which corresponds to the exponent at Y of the monomer considered.

  As previously for the value A, it is verified in step E519 that the value B is not strictly less than the multiplicity m (i) for the index point i minus the current value A of the monomer exponent. Considered in X. If the exponent B is much smaller than [m (i) -A], processing is continued for the index point i at step E521. In the opposite case, the processing of all the exponents B according to the variable Y that can be envisaged for the value A having been carried out, is passed to the following value A in the step E520 (by incrementing the value A). Once the value A has been incremented, we return to the step E516 previously described.

  Processing continues in step E521 by initialization to 0 of the index L which, as we have seen previously, successively designates each candidate polynomial, and the initialization to + oo of a minWd variable used in following (in practice, it is naturally an initialization with a very important value, for example the largest integer value that can take the corresponding register).

  We will now describe a loop on all candidate polynomials.

  Indeed, the step E521 is followed by a step E522 to which one verifies whether the index L has reached the number of candidate polynomials T. If all the candidate polynomials ZL have been processed, one leaves the loop at the step E526 described later. If, on the other hand, the index L still denotes a polynomial to be processed, proceed to step E502.

  In step E502, the coefficient D of the exponent monomer A at X and B at Y is determined in the translated polynomial ZL (X + x ;, Y + y;).

  We then go to step E523 where we check if the coefficient D is zero. If this is the case, go to step E525 where the index L is incremented so as to consider the next candidate polynomial, then to step E522 where it is checked whether the preceding candidate polynomial was not the last to consider. Step E522 has been described above.

  If the step E523 determines that the coefficient D is not zero, step E524 is carried out, in which the weighted degree of the candidate polynomial ZL is compared to the minimum minWd of the weighted degrees obtained hitherto.

  If the weighted degree of the current polynomial ZL is greater than or equal to the minimum already obtained minWd, proceed to the next candidate polynomial by proceeding to the step of incrementing the index L E525 previously described.

  If, on the contrary, the weighted weight of the current polynomial ZL is strictly less than the minimum already obtained minWd, then the current polynomial ZL is the new polynomial of least weighted degree. In this case, therefore, step E503 for storing values relative to the current polynomial ZL is carried out.

  Indeed, in step E503, the current polynomial ZL is stored as the polynomial Zm; n. The minimum weighted degree minWd is also updated to the weighted weight value of the current polynomial ZL. The coefficient D is also memorized as the coefficient relative to the minimum weighted weight, denoted minD. Finally, the index L of the current candidate polynomial ZL is stored under the value minL.

  The data of the current polynomial ZL having been stored as those of the minimum weighted weight polynomial Zmin, it is possible to pass to the next candidate polynomial by proceeding to step E525 for incrementing the index L previously described.

  As seen previously, when the set of candidate polynomials ZL has been processed in step E502, the processing loop is released in step E522 to proceed to step E526.

  In step E526, the index L, which designates the current candidate polynomial, is initialized again.

  We will now perform a new loop on all candidate polynomials.

  To do this, we proceed to step E527 to verify that the current polynomial L must be processed, that is to say that the index L is much lower than the total number of candidate polynomials T. If it is not is not the case, we leave the loop to go to step E530 described below. On the other hand, if the index L represents a polynomial to be processed, proceed to step E528 where it is determined whether the current index L is equal to the minL index of the minimum degree polynomial, such as it was determined by successively following step E503 previously described.

  If the index L corresponds to the minL index of the minimum degree polynomial, proceed to step E504b where the corresponding polynomial Zmin is multiplied by the polynomial X + xi. The next candidate polynomial is then incremented by the index L in step E529.

  If in step E528 it was determined that the index L of the current polynomial 25 was different from the minL index of the minimum weighted degree polynomial, step E504 is carried out at which the current polynomial ZL is modified by replacing it by the following polynomial: ZL.minD + D.Zmin. We then also go to step E529 of incrementation of the index L, followed by step E527 previously described.

  When the processing of the steps E504a or 504b has been performed for all the candidate polynomials ZL, the step E527 leads to the step E530 at which the exponent B is incremented according to the variable Y. Once the exponent B incremented in step E530, we return to the step E519 described above where it is determined whether all the exponents B to be treated were it.

  When all the points have been treated by the passage of the index i from the value 1 to the value, and for each point, the set of exponents A, B that can be envisaged for the variables X and Y have been treated, the step E514 conducted as already mentioned in step E505.

  In step E505, the polynomial Q (X, Y) is chosen which results from the set of the algorithm described as being the candidate polynomial ZL (X, Y) of minimum weighted degree, and in particular to simplify the calculations. who will follow. If several candidate polynomials ZL have the same weighted degree, we consider in the following the plurality of candidate polynomials ZL of the same degree.

  Step E505 therefore ends the interpolation algorithm.

  The interpolation phase thus generates at least one Q polynomial which vanishes in all points (x;, ri (x,)), for i = l, u.

  In the case presented as an example here, we obtain in the treatment of r1 the polynomial Q1 (X, Y) = a12X + aX4 + a9X5 + aX6 + a6X7 + a3X8 + a4X10, + a5X11 + a13Y + aX2Y + a10X3 + a12XY, and in the treatment of r2 the polynomial Q2 (X, Y) = a12Y + a11XY1-a3X2Y +1 + a11X + a5X2 + a6X4 +0 (, 2X5 + a5X6 + a10X7 + a5X8 + a8X3 + a3X9 + a2X10 + a14X11 + a14X12 The processing of the aggregated word ri then continues with a factorization phase of the Q polynomial (s) obtained previously in order to obtain at least one factor of the form [Yf (X)].

  The factorization phase is for example carried out in accordance with an algorithm of the Roth-Ruckenstein type, as described in FIG. 6. For more details on this algorithm, one can refer to the article Efficient Decoding of Reed Solomon Codes Beyond Half the Minimum Distance, Roth R. and Ruckenstein G., IEEE Trans. Inform. Theory, vol. 46, No. 1, January 2000.

  The factorization algorithm starts with a step E601 to which the polynomial Q (X, Y) is stored for its processing as a polynomial M (X, Y).

  The factorization algorithm is a recursive algorithm. An index i indicates the number of loops performed within this algorithm. Note also v; the number of roots found during the index loop i of the algorithm.

  Thus, in step E610, the index i to zero is initialized, as is the number vo of roots found during the loop of index O. The algorithm can then start in the proper sense at the step E611 to which it is initialized. to zero the value of an index r.

  Next, step E612 is made to determine if the polynomial Xr divides the current M (X, Y) polynomial. If this is the case, we will investigate whether a higher power of X also divides the polynomial M (X, Y). To do this, we go to step E613 where the variable r is incremented, then we return to step E612 which has just been described.

  If it is determined in step E612 that the polynomial Xr does not divide the current polynomial M (X, Y), we proceed to step E602 where the current polynomial M (X, Y) is divided by the polynomial X This division is of course possible given the determination made during the penultimate pass at step E612.

  We then go to step E615 to which a variable t is initialized to zero, which will make it possible to successively designate all the elements of the Galois body Fq considered.

  Indeed, in step E616 following step E615, it is determined whether the value of the polynomial M (X, Y) for X = at and Y = 0 is zero. If this is not the case, proceed to step E603 described below. If on the other hand it is the case, one proceeds to step E621 of incrementation of the variable t, then to the step E622 where one verifies if t reaches the number of elements of the Galois body Fq. If not, return to step E616 described above; in the affirmative, the polynomial M (X, Y) being canceled on all the values of X of the Galois field for Y = O, the algorithm is finished and we thus go to the step E606a which concludes it.

  Indeed, the step E606a consists of defining the polynomial f (X) according to the results of the algorithm: f (X) = go + g1X + ... + gk_iXk-1.

  Returning to step E603 previously mentioned, an exhaustive search is made among the elements of the Galois field Fq of the Nr roots of the polynomial M (0, Y), which will be called g ,,;

  We then go to step E617 to which we determine if the index v; roots found during the index loop i reaches the number of roots Nr. If this is the case, we can proceed to step E618 of incrementation of the index of loop i, that is to say say to the next loop in the sense of the recursive algorithm. It is noted that the number of roots v is also initialized in step E618; from the new loop to zero. After step E618, the execution of the new loop is started in step E611 described above.

  If in step E617 it is determined that the index of the root v; is less than or equal to the number of roots Nr, proceed to step E619 at which it is determined whether the loop index i has reached the dimension k of the code, related to the degree k-1 of the polynomial f (X) sought.

  If this is the case, that is to say if i = k, we go to step E606b at which it is determined that the algorithm did not find a polynomial f (X) solution. This step naturally leads to the failure of decoding for the polynomial Q (X, Y) to which the factorization algorithm has been applied.

  If it is determined in step E619 that the loop index i has not reached the dimension of the code k, we go to the step E604 where the current polynomial M (X, Y) is transposed by the definition following of the new current polynomial: M (X, Y) = M (X, Y + g;).

  Step E604 is followed by step E605 to which the polynomial M (X, Y) is shifted: as previously the current polynomial is replaced by the new current polynomial according to the formula: M (X, Y) = M ( X, XY).

  After step E605, proceed to step E620 to which the index v is incremented; which denotes a particular root found during the loop i. Step E620 is followed by step E611 previously described which allows the continuation of the algorithm.

  The algorithm which has just been described has thus made it possible to determine a polynomial f (X), defined in step E606a, such that [Y-f (X)] is a factor of the polynomial Q (X, Y).

  2869482 33 Since the polynomial [Y-f (X)] is a factor of Q, it vanishes in at least a part of the points (x ,, r] (x,)), for i = 1, u. At the points (x ,, rj (xi)) where [Y- f (X)] vanishes, we can write r.) (X;) = f (x,) and the aggregated word rj therefore has as many common components with the word consisting of the components f (x,) for i = 1, u.

  The aggregate word r such that ij (x,) = f (x,) for i = l, u, which is a code mode in the considered Reed-Solomon code, is therefore a good candidate for being an associated aggregated word to the received word aggregated r.

  The application of the factorization algorithm which has just been described to the polynomial Q1 (X, Y) obtained above in relation to the aggregate received word r1 gives as polynomial factor Q1 (X, Y) the following polynomial : (Y + a5X9 + a 10X8 + a4X7 + a 12X6 + aX5 + a3X4 + aX3 + a13X2 + a 14X) The application of the same algorithm to the polynomial Q2 (X, Y) obtained from the aggregated received word r2 gives as a factor the polynomial: (Y + a11X10 + a13X9 + a4X8 + a8X7 + a7X6 + a11 X5 + 068X4 + a13X3 + a13X + (13) .These polynomials are actually of the form [Yf (X)], we obtain the associated aggregated code word to each one by calculating for each, as indicated above, the value taken by the polynomial f on the points of the set of localization (x ,, ..., x,). One can thus consider that this step consists of re-encoding the word.

  The associated aggregated codewords are thus obtained: = (a, 064, 062, a1, CL12, a9, a0, 0, 0,0, o, 0,0,0,0) and r2 = (a13, 2, a9 2a13, a11, a5, 0 0, 0, 0 0, 0 0,0), where the bold numbers highlight the components that have been modified, that is, corrected, with respect to aggregated received words r1 and [2.

  Each zone 310,..., 320 thus makes it possible to obtain, according to a list decoding method, an associated aggregated codeword r i from the aggregated word r.

  Aggregated words v; corresponding to the words v of the algebraic geometry code by the general relation: Â (x) -1 vj (x) E [yp (x)] 3 v (x, .yp (x)), p = o one will be able to determine the components of the associated codeword r to the received word r by solving the system composed of the following equations: (x;) = E [y, (x,)} '(xi, Yp (x;)) (for i = 1, u and j = 0, ..., max), p = 0 since the aggregated associated words r are now known.

  The resolution of the system of equations, schematically represented by the block referenced 350 which receives the set of associated aggregated words rj of the different zones 310,..., 320, thus makes it possible to obtain the codeword r associated with the received word r.

  In the example presented in detail here, the resolution of the system results in the following associated code mode: (a9, a12 a13 a5 a127 a9, a, 0, 0, 0, 0, 0, 0, a7, a6, a14, a2, 0, 0, 0, 0, 0, 0, 0, 0), equal to the code word initially sent.

  The error correction by means of the list decoding of each of the aggregated received words has therefore been effective.

  It may be noted that it is possible alternatively to solve in the system of equations only those lines which correspond to erroneous aggregated received words, that is to say for which ij (x;) r; (Xi). The computational cost will thus be limited since the number of erroneous components is generally low.

  FIG. 7 schematically shows a possible solution in the case where the list decoding of the aggregated received words ri gives several solutions, for example two polynomials f1 and f2 which correspond to two solutions &, o and il;; each received word aggregated r; In the processing of each aggregate received word r; (that is to say in each zone 310, ..., 320), a multiplexer 720a, ..., 720b makes it possible to transmit to the re-encoding device 730a, ..., 730b only one polynomials f1, f2 obtained by the factorization algorithm and stored in memory (as symbolically represented with references 710a, ..., 710b).

  The polynomials selected by the multiplexers 720a,..., 720b thus generate associated aggregated words which are used in the resolution of the system represented as previously by the reference 350, in order to obtain an associated code word. For each possible selection between different polynomials f2 performed by the multiplexers 720a,..., 720b, an associated word r is obtained which can be compared with the received word r (referenced 760 in FIG. 7) within a comparator device. 360.

  By controlling all the possible combinations as to the selections of the various multiplexers, the comparator 360 can thus determine which combination makes it possible to obtain an associated word F closest to the received word r in the Hamming distance sense, and to choose this associated word. as a result of decoding.

  It will be possible to use without departing from the scope of the invention other methods of choice of the decoding result when the list decoding method gives a plurality of solutions.

Claims (34)

  A decoding method for a code of length n and of dimension k on a Galois field Fq, the code being defined by means of a cardinalization set n on Fq x Fq, characterized in that, the set of localization comprising at least fu subsets called aggregates, where, u 2, each aggregate grouping the elements (x, yp (x)) of the localization set having a common value of x, for p = 0, ... , 2 (x) 1, where,. (x) is the cardinal of the aggregate, it comprises the following steps: computation (315a, 315b), from a received word r, of aggregated received words r, according to said, u aggregates; for each aggregated received word r1, obtaining (100) at least one associated aggregated word iL-j in a Reed-Solomon code by a list decoding method; calculating (350) at least one associated word i, in the code of length n and of dimension k, whose corresponding aggregated words according to said, u aggregates are the aggregated associated words i ^ J. 2. Decoding method according to claim 1, characterized in that the code is defined by a parity matrix H whose elements Ha, p for 1 <_ / 3 n are equal to the value taken by a monomer ha = Y ' X 'on the points of the location set, the monomials ha for 1 sank all having a distinct weight.   3. Decoding method according to claim 1 or 2, characterized in that the code is a code of algebraic geometry.   2869482 37 4. Decoding method according to one of claims 1 to 3, characterized in that the calculation of an aggregate received word ri = [5 (x1), r1 (x2),   ., r1 (xi,)] is realized according to a formula of the type ri (x) = A (x) [r (x, y0 (x)), ..., r (x, y? (x) _, (x))], where Ai (x) is a function on Fei for all j between 0 and jmax, where jmax 1 ... CLMF: 5. decoding method according to claim 4, characterized in that, for all i between 1 and, u and for all j between 0 and Lm, the function A (x,) 2 (x,) - 1 is defined by AJ (x,) [zo, ..., z2 (x,) _, 1 = E [Yp (x,) fzp. p = 0   6. Decoding method according to claim 4 or 5, characterized in that the components i (x, y) of the associated code word are such that (x,) [(x ya (x,)), ..., i - (x,, yz (x,) _, (x,))] = (x,) for all i between 1 and, u and for all j between 0 and jmax 7. decoding method according to one of claims 1 to 6, characterized in that the associated word is calculated by the resolution of a system of equations.   8. Decoding method according to claim 7, characterized in that the resolution of the system of equations is limited to the lines which contain erroneous components of the aggregated received words r, for which rf (x;) #ri (x,). 9. decoding method according to one of claims 1 to 8, characterized in that it comprises a step of selecting an associated word from a plurality of calculated associated words.   10. Decoding method according to one of claims 1 to 8, characterized in that the list decoding method comprises, for each aggregate received word ri, the following steps: - search for at least one polynomial fi on Fq of degree at most ki 5 such that the number of components ri (x) of the aggregate received word ri for which ri (x) = f1 (x) is at least equal to a number where ki is a predetermined integer and where ris <p and; calculating the associated aggregated word r such that (x1) = f (x,) for all i between 1 and p. 11. Decoding method according to claim 10, characterized in that the number is predetermined.   12. A decoding method according to claim 10, characterized in that the number is the maximum number of components for which r (x) = gJ (x) when g traverses the set of polynomials over Fq of degree at most ki.   13. Decoding method according to one of claims 10 to 12, characterized in that, the step of searching for at least one polynomial f generating at least two polynomials solutions, it comprises the following steps: - for each polynomial solution, calculation of an associated aggregated word; for each solution polynomial, calculating at least one associated word as a function of said associated aggregated word calculated for the solution polynomial; comparing the associated words calculated for each solution polynomial with the received word r; selecting the associated word among the associated words calculated for each solution polynomial.   14. Decoding method according to claim 13, characterized in that the selected associated word is the closest associated calculated word to the received word r in the sense of the Hamming distance.   15. A decoding method for a code of length n and of dimension k on a Galois field Fq, the code being defined by means of a cardinalization set n on Fq x Fq, characterized in that, the set of localization comprising at least u subsets called aggregates, where u 2, each aggregate grouping the elements (x, yp (x)) of the localization set having a common value of x, for p = 0, ... , 2 (x) -1, where / t (x) is the cardinal of the aggregate, it comprises the following steps: computation, from a received word r, of aggregate received words ri = [ri (xl) , ri (x2), ..., ri (xp)] for j = 0, ..., jmax, where,% max? 1, according to the formula  (x) -I ri (x) = E [yp (x)] 'r (x, yp (x)), p = o - for each aggregated received word ri, search for at least one polynomial fi over Fq of degree at most k. such that the number of components r (x) of the aggregated received word r for which r (x) = f (x) is at least equal to a number ris, where i, u; for each received aggregate word ri, calculating at least one associated aggregated word r such that if (x,) = fj (x,) for all i between 1 and u; calculating at least one associated codeword r whose components 1 (x) -I i, (x, y) are such that i, (x1) = E [yp (x1)] 'ï "(x, , yp (x,)) for all i between 1 and p = 0 u and for all j between 0 and Lm.   16. A decoding device for a code of length n and of dimension k on a Galois field Fq, the code being defined by means of a cardinalization set n on Fq x Fq, characterized in that, the set of localization comprising at least, u subsets called aggregates, where, u 2, each aggregate grouping the elements (x, yp (x)) of the localization set having a common value of x, for p = 0 ,. .., 2 (x) l, where Z (x) is the cardinal of the aggregate, it comprises: - means of calculation, from a received word r, of aggregated received words r according to said, u aggregates ; means for obtaining, for each aggregated received word r, at least one associated aggregated word rJ in a Reed-Solomon code by a list decoding method; means for calculating at least one associated word II ", in the code of length n and of dimension k, whose corresponding aggregated words according to said, u aggregates are the associated aggregated words r.   17. A decoding device according to claim 16, characterized in that the code is defined by a parity matrix H whose elements Ha, a for 1 <Q <_ n are equal to the value taken by a monomer ha = Y 'X' on the points of the localization set, the monomials ha for 1 a <nk all having a distinct weight.   18. Decoding device according to claim 16 or 17, characterized in that the code is a code of algebraic geometry.   19. Decoding device according to one of claims 16 to 18, characterized in that the means for calculating an aggregate received word r1 = [ri (x1), r1 (x2), ..., r] ( xp)] incorporate calculation means of the type r (x) = Ai (x) [r (x, yo (x)), ..., r (x, y, (x) _1 (x))], where A; (x) is a function on Fe) for all j between 0 and jmax, where Imax 2_1.   20. The decoding device as claimed in claim 19, characterized in that for all i between 1 and p and for all j between 0 and j, nax, the function% (x;) 1 Aj (xi) is defined by A1 ( xi) [z0, ..., za, (x) _1] = E [yp (x1)] 'zp p = 0 21. Decoding device according to claim 19 or 20, characterized in that the means for calculating the associated codeword ï ^ determine the components i, -. (X, y) of the associated code word i such that Aj (x;) [r (x ;, yo (x;)), ..., r (x ,, y? (x)? I (x;))] = i'j (x;) for all i between 1 and p and for all j between 0 and jmax 22. The decoding device according to one of claims 16 to 21, characterized in that the means for calculating the associated codeword include means for solving a system of equations.   23. A decoding device according to claim 22, characterized in that the resolution means of the system of equations are limited to the processing of the lines which contain erroneous components of the aggregated received words r, for which (x;) ri (x; ). 24. decoding device according to one of claims 16 to 23, characterized in that it comprises means for selecting an associated word from a plurality of calculated associated words.   25. Decoding device according to one of claims 16 to 23, characterized in that the means for obtaining the associated aggregated word L i by a list decoding method include: - search means of at least a polynomial fi over Fq of degree at most kf such that the number of components rf (x) of the aggregate received word rf for which rf (x) = fi (x) is at least equal to a number; where kf is a predetermined integer and where 77 f _ <, u and; means for calculating the associated aggregated word j such that (x,) = fi (x,) for any i between 1 and, u.   26. The decoding device as claimed in claim 25, wherein the means for searching for at least one polynomial generating at least two solution polynomials, it comprises: means for calculating an associated aggregated word for each polynomial solution,; calculating means, for each solution polynomial, of at least one associated word as a function of said associated aggregated word calculated for the solution polynomial; means for comparing the associated words calculated for each solution polynomial with the received word r; means for selecting the associated word r from the associated words calculated for each solution polynomial.   27. A decoding device for a code of length n and of dimension k on a Galois field Fq, the code being defined by means of a cardinalization set n on Fq x Fq, characterized in that, the set of a location comprising at least, u subsets called aggregates, where, u 2, each aggregate grouping the elements (x, yp (x)) of the location set having a common value of x, for p = 0, (x ) -1, where 2 (x) is the cardinal of the aggregate, it comprises: - means of calculation, from a received word r, of aggregate received words r = [r (x1), ri (x2 ), ..., ri (x,)] for j = 0, ..., jmax, where .max? 1, according to the 2 (x) - 1 formula ri (x) _ E [yp (x)} r (x, yp (x)), p = o - search means, for each aggregated received word rj, of at least one polynomial fj over Fq of degree at most ki such that the number of components ri (x ) of the aggregated received word r for which rj (x) = fj (x) is at least 77; , where s u; calculating means, for each aggregate received word rj, of at least one associated aggregated word r such that (x,) = fj (x1) for all i between 1 and, u 10 - calculating means from minus one associated codeword whose 2 (x) 1 the components i, (x, y) are such that t "j (x1) = [yp (x,)] r (x1, yp (x1)) for p = o all i between 1 and, u and for all j between 0 and jm.   An encoded digital signal reading apparatus comprising a modulated data reader, means for demodulating said modulated data into coded digital signals, and a decoder device according to one of claims 16 to 27 for decoding said encoded digital signals.   An encoded digital signal receiving apparatus comprising a modulated data receiver, means for demodulating said coded digital signal modulated data, and a decoder device according to one of claims 16 to 27 for decoding said encoded digital signals.   Data storage system comprising at least one coded digital signal recording apparatus, at least one recording medium and at least one coded digital signal reading apparatus according to claim 28.   A data communication system in the form of blocks of predetermined length, comprising at least one digital signal transmitting apparatus and at least one coded digital signal receiving apparatus according to claim 29.   32. Non-removable data storage medium comprising computer program code instructions for performing the steps of a decoding method according to one of claims 1 to 15.   33. Partially or fully removable data storage means comprising computer program code instructions for performing the steps of a decoding method according to one of claims 1 to 15.   34. Computer program, containing instructions such that, when said program controls a programmable data processing device, said instructions cause said data processing device to implement a decoding method according to one of claims 1 to 15.