Abstract: The present invention concerns an encoding method in which encoding is performed of any information word a of length k in the form of a word ? belonging to a Reed-Solomon code C of dimension k? and length n? (with n??k?=n?k) such that the components of ?? situated in (n??n) arbitrary predetermined positions be systematically equal to respective predetermined constants (for example, all zero). The possibility then exists of deleting those components of fixed value to obtain a word ? of length n belonging to a code C, which thus constitutes a code that is shortened with respect to code C. The invention also relates to devices and apparatuses adapted to implement the encoding method. The invention may be used for encoding by means of an algebraic geometric code, when such encoding may be implemented by encoding by means of a plurality of shortened Reed-Solomon codes.

Abstract: A method of decoding product codes is disclosed, in which the symbols of each codeword may be placed in a table comprising n2 rows and n1 columns, such that the symbols constituting each row form a permitted word of length n1 according to a first component code able to be decoded by means of an algorithm A1 for correction with erasures, and the symbols constituting each column form a permitted word of length n2 according to a second component code able to be decoded by means of an algorithm A2 for correction with erasures. According to the method, a correction of a row or column is only accepted when the result of the correction is deemed reliable, otherwise all the symbols of that row or column are erased. Devices and apparatus adapted to implement this method are also disclosed. The method is preferably applied to algebraic geometric codes.

Abstract: To interleave a binary sequence a represented by the polynomial a ? ( x ) = ? i = 0 n - 1 ? a i ? x i , where n=R.M with R?M, i being an integer which may be written i=r.M+c, r and c being integers, r?0 and c ? [0, M?1], there is obtained, from the sequence a, an interleaved binary sequence a*. The interleaved binary data sequence a* represented by the polynomial a * ? ( x ) = ? i = 0 n - 1 ? a i ? x i * where i*=[r?h(c)].M+c mod n, the h(c) being obtained by the choice of an M-tuple h0=[h0(0), . . . , h0(M?1)] of non-negative integers less than R?1 such that, given a predetermined set ? of circulating matrices P of dimension M×M, for any matrix P of ?, the residues modulo R of the components of the vector h0.P are not nil; and the corresponding choice of an M-tuple h obtained from h0 by the application of a permutation moving h0(c) to position L×c mod M, the integer L being relatively prime with M.

Abstract: A method for the encoding of a sequence of source words, the method comprising: a step for the selection of an encoding function to be applied to a source word to be encoded as a function of the content of said word to be encoded and of at least one preceding source word in said sequence so that the concatenation of two consecutive encoded words has no binary element, called an isolated binary element, sandwiched between two binary elements with a value different from that of said isolated binary element; and a step for the encoding of said word to be encoded implementing the selected encoding function.

Abstract: Turbocoding methods use a first RSC coder operating on sequences of binary data a, and a second RSC coder operating on binary sequences a* each obtained from a by means of a predetermined permutation. These permutations are designed so that, for any sequence a represented by a polynomial divisible by the recursion polynomial, the associated sequence a* is also represented by a polynomial divisible by the recursion polynomial. These permutations are relatively simple to implement, and are applicable to all the data sequences a which have a length which is a multiple of the period of the recursion polynomial. In addition, once the transfer functions of the coders and the sequence length have been chosen, it is possible to select, amongst the corresponding permutations, the one which will probably offer the highest minimum distance of the code. Application to devices and apparatus implementing these methods.

Abstract: The present invention concerns a method of decoding a one-point algebraic geometric code defined on an algebraic curve represented by an equation in X and Z of degree 2?? in Z, where ? is a strictly positive integer and ? an integer greater than 1, obtained by taking the fiber product of ? component algebraic equations, each of said component equations governing the unknown X and an unknown Yi, where i=0, . . . , ??1, and being of degree 2? in Yi. This method comprises the decoding of 2(??1)? “clustered” codes, all defined on the same algebraic curve represented by one of said component equations. The invention also relates to devices and apparatuses adapted to implement this method.

Abstract: A method of encoding information symbols comprises a step in which a word v, orthogonal to a matrix H, the element H?? of which is equal to the value taken by some monomial h?=YjXi at the point P? of some locating set, is associated with every block of k information symbols belonging to a Galois field Fq. The invention shows how to choose the set of said monomials h? so as to define codes which can be decoded with an algorithm by aggregates of low complexity, and which provides a very good error correction capability, in particular for channels in which the errors tend to occur in bursts. The invention also relates to devices and apparatuses adapted to implement this method.

Abstract: A method of decoding a one-point algebraic geometric code defined on an algebraic curve of the kind C(a,b), represented by an equation of degree b in X and of degree a in Y, comprises, for any received word, a step of locating transmission errors affecting said received word. The correction of errors in said word, which belongs to an algebraic geometric code, is then reduced to the correction of errors in a certain number, at most equal to a, of words belonging to a Reed-Solomon code. Devices and apparatuses adapted to implement this method are also described.

Abstract: In a method of decoding a one-point algebraic geometric code of dimension k and length n, in order to identify the position of the errors in a received word, the syndromes matrix S, of dimension (n?k)×(n?k) is defined, of which the elements Sij of each line i are calculated, for j between 1 and w(i), where the boundary w is a decreasing function, using the syndrome s of the received word, as well as the matrix S* obtained by “extending” the matrix S, that is to say by calculating the value of certain elements S*ij where j is greater than w(i). This method makes it possible in certain favorable cases to find the erroneous positions of the received word when the number of errors is greater than (n?k+1?g)/2, even if it is not possible to calculate all the elements of S* conventionally required by a two-stage algorithm to perform that correction. Devices and apparatuses adapted to implement this method are also discussed.

Abstract: The present invention concerns a device (10) for the encoding of information symbols to transmit or to record, and for the correction of errors among the symbols received or read, according to codes defined over a Galois field Fq, where q is an integer greater than 2 and equal to a power of a prime number, and in which a set of elements of Fq are considered which are denoted yl(j), where j=1, . . . , R with 1?R?q?1 and l=0, . . . , p?1 with p>1.

Abstract: In order to decode a sequence ?=(?1, . . . , ?i, . . . , ?n) where ?i is the received electrical signal corresponding to a transmitted signal ai representing the ith binary element vi of a word v=(v1, . . . , vn) chosen in a code C of words satisfying v·hT=0, where h is a row n-tuplet on the set {0,1 }, whose number of 1 is denoted w, an item of extrinsic information ?ext[A(i,h)]=P[ai=?1|A(i,h)]/P[ai=+1|A(i,h)] is determined on each of the elements vi covered by h, A(i,h) being the set of the received values ?j covered by h, with the exception of ?i, and P[ai|A(i,h)] being the probability that the ith signal transmitted was ai. This gives ?ext[A(i,h)]=[S1(i)+S3(i)+ . . . ]/[1+S2(i)+S4(i)+ . . .

Abstract: The present invention concerns a method of coding information symbols according to a code defined on a Galois field Fq, where q is an integer greater than 2 and equal to a power of a prime number, and of length n=p(q?1), where p>1. This coding is designed so that there exists a corresponding decoding method, also disclosed by the invention, in which the correction of transmission errors essentially comes down to the correction of errors in p words of length (q?1) coded according to Reed-Solomon. The invention is particularly advantageous when, through a suitable choice of parameters, the code according to the invention is an algebraic geometric code: in this case, it is possible to correct the transmission errors by the method already mentioned and/or by a conventional method which is less economical but has a higher performance. The invention also concerns devices and apparatus intended to implement these coding and decoding methods.

Abstract: In order to encode an original sequence of binary data (u), a first padding operation (508) is performed, supplementing the original sequence (u) so that the supplemented sequence (u) is divisible by a first divisor polynomial; a first recursive convolutional encoding operation (508) is performed, using the first divisor polynomial, encoding the supplemented original sequence (u); an interleaving operation (506) is performed, permuting the binary data in the original sequence (u) by means of a specific permutation, so as to obtain an interleaved sequence (u*); a second padding operation (510) is performed, supplementing the interleaved sequence (u*) so that the supplemented interleaved sequence (u*) is divisible by a second divisor polynomial (g2); and a second recursive convolutional encoding operation (510) is performed, using the second divisor polynomial, encoding the supplemented interleaved sequence (u*).

Abstract: The present invention concerns channel codes particularly well adapted to transmission in channels in,which errors tend to occur in bursts. Moreover, the codes according to one embodiment of the invention using an algebraic geometric curve are easy to decode and have a relatively high minimum distance. The invention also relates to the corresponding encoding and decoding methods, as well as the devices and apparatuses adapted to implement those methods. Application is in particular to mass storage, and to systems of communication by OFDM.

Abstract: The coding method to which the present invention relates takes into account at least one selection criterion related to a transmission of binary symbols representing a physical quantity, for selecting at least one transmission parameter. Each said selected transmission parameter is in the set of parameters comprising: a number K, greater than or equal to 1, of sequences ai (i=1, . . . , K) of binary symbols, to be coded, an integer M1, equal to or greater than 2, a divisor polynomial gi(x), an integer M, an interleaver, and a multiplier polynomial fij(x). It uses an interleaver which maintains the divisibility by a polynomial gi(x).

Abstract: The present invention relates to a A method of decoding a one-point algebraic geometric code of dimension k and length n, in which, in order to identify the position of the errors in a received word, the syndromes matrix S, of size (n−k)×(n−k), is defined, of which the elements Sij of each line i are calculated, for j between 1 and w(i), where the boundary w is a decreasing function, using the syndrome s of the received word. According to the invention, matrices Matrices Su are constructed for the successive values of u starting with S1=S, and, for u>1, each matrix Su is obtained by performing on the matrix Su−1, column permutations where appropriate, then linear manipulations involving the line of index u. These steps are performed in such a manner as to find a matrix S&lgr; which has a line of index less than or equal to &lgr; of which the elements are zero in the first w(&lgr;) columns. The invention also relates to devices and apparatuses adapted to implement this method.

Abstract: A method of decoding product codes is disclosed, in which the symbols of each codeword may be placed in a table comprising n2 rows and n1 columns, such that the symbols constituting each row form a permitted word of length n1 according to a first component code able to be decoded by means of an algorithm A1 for correction with erasures, and the symbols constituting each column form a permitted word of length n2 according to a second component code able to be decoded by means of an algorithm A2 for correction with erasures. According to the method, a correction of a row or column is only accepted when the result of the correction is deemed reliable, otherwise all the symbols of that row or column are erased. Devices and apparatus adapted to implement this method are also disclosed. The method is preferably applied to algebraic geometric codes.

Abstract: The coding method to which the present invention relates takes into account: a polynomial without square g(x), N0, the smallest integer such that g(x) is a divisor of the polynomial xN0+1; n, an odd multiple of N0; a sequence u of n symbols ui to be coded; and e, a power of 2 different from 1, which is residue of e modulo N0 is equal to 1. It includes: an operation of forming a concatenated sequence uu* (304) consisting successively of the sequence u and a sequence of symbols u* such that u*(x)=u(xe) modulo xn+1, an operation of coding the concatenated sequence uu* (305), including at least one division of the concatenated sequence uu* by the polynomial g(x).