WO2015107315A1 - Encoding and decoding linear code from lattices - Google Patents

Abstract

The invention relates to an encoding and decoding method using a linear error-correcting code (n,k), which comprises a computation phase with a plurality of stages and, for each stage, a computation step implementing a butterfly operator: in order to determine T _i ⊗T _j products of separate elementary lattices, each elementary lattice being obtained from one of the k lines of the generator matrix or one of the n-k lines of the parity check matrix; in order to compute, on a T _i ⊗T _j product lattice, a posteriori probabilities of the states of the T _i ⊗T _j product, knowing the probabilities of the states of the elementary lattices; and in order to calculate a posteriori probabilities of the states of each of the two lattices T /. and T. knowing the a posteriori probabilities of the states of the T _i ⊗T _j product lattice, the method comprising a phase of determining a posteriori probabilities associated with the symbols decoded from the a posteriori probabilities of the states of the k or n-k elementary lattices of the last stage.

Description

 CODING AND DECODING LINEAR CODE FROM TREILLIS

Field of the invention

 The field of the invention is that of coding and decoding of symbols, implementing an error correction code.

 The invention relates more particularly to source symbol coding techniques, delivering coded symbol packets or a coded symbol stream comprising the source symbols and redundancy symbols, intended to be transmitted over a transmission channel ( for example overhead or wired of wireless type, optical or electrical), or stored (s) in a material medium. The invention also relates to corresponding decoding techniques, in particular for correcting the transmission errors inherent in the transmission channel.

 The invention finds particular applications in the following fields:

 the transmission of information by wired electrical telecommunications, as in the ADSL or optical standards, on optical fibers or in free space;

 the transmission of information in space and wireless terrestrial radio communications ("wireless"), as in digital television, DAB digital radio, GSM or UMTS telephony, WiFi radio network, and also in future telecommunications systems, such as future standards for television, radio broadcasting, voice, video and data applications (DVB, LTE, 4G, 5G, etc.), "Future Internet", or between vehicles, communicating objects or machines ...;

 information source compression and decompression;

 the generation and detection of so-called scrambling sequences in CDMA systems;

 storing information in magnetic, optical, mechanical or electrical mass memories for constituting hard disks, or RAMs of computers, or memory sticks with a USB type interface ...;

 correcting information during calculations in an integrated circuit of a microprocessor or in a computer;

 pattern recognition: images, sounds, etc.

 robotics controlled by an artificial intelligence based on neural networks. Prior art

 Many coding techniques allow the correction of decoding transmission errors by generating redundancy symbols from source symbols.

 Thus, an error correction code is conventionally defined by:

a length n, corresponding to the symbols at the output of the encoder (length codeword n formed of k source symbols and (n - k) redundancy symbols),

 a number of bits or useful information symbols k corresponding to the input symbols of the encoder, also called source symbols, and

 a minimum distance d ^.

The minimum distance of a code d _m i _n is the minimum distance between two codewords. It allows to determine the maximum number of errors that the code can correct in a code word.

Thus, the greater the minimum distance dmin is, the better the error correcting code, since it allows to detect (d _m i _n - l) erroneous symbols and correct in [(ίίηύη ^- J ^ ° ^u the operator means the whole part).

 Calderbank-Forney-Vardyfl] described the construction of elementary lattices. The elementary lattices are constructed using the lines of either a generator matrix G or a parity check matrix H of the block code.

A block code C (n, k) is defined by a generative matrix G of dimensions kxn composed of k rows and n columns such that for any information block x = χ ₀ , Χχ, ...

the code word c = (c ₀ , c _lt ..., c _n-1 ), obtained by coding, that is to say the vector-matrix product of x and G is given by:

 c = x. G (mod 2)

where the matrix G is such that:

The lines g _t of the matrix G are called generators and the code word c is thus obtained by making the weighted sum of the g; by the useful information bits Xj:

 fc-l

c = ^ x _t 9i

 £ = 0

Each line g _t generates a subcode of C denoted by C _g Let Ti be the trellis corresponding to the code C _i, then the global trellis T representing the code C is obtained by the product of all the k elementary trellises:

The lattices T _{ are elementary lattices with two states. Each trellis represents two code words: the code word zero corresponding to the information bit x _t = 0 and the code word g _j corresponding to the information bit x _t = 1. Each elementary trellis is therefore composed of two paths. representing these two codewords. In the case of a code generating matrix G, the construction of an elementary lattice and the product of two lattices are described from the example of the Hamming code (n = 7, k = 4, d _m j _n = 3) whose generator matrix G _4x7 is given by:

 Ό 1 1 0 1 0 0>

 1 0 1 1 0 0 0

 ¾X7- 1 0 1 0 1 1 0

 α ο ο ο ι ο

An elementary lattice T _j represents the ith line in the generator matrix G of the code. FIG. 1 represents the elementary lattice 7 ^ corresponding to the generator gi in the case of a generator matrix. For the preceding Hamming code (7, 4, 3), the two elementary lattices associated with the first two rows of its generating matrix above are represented in FIGS. 2 and 3. FIG. 2 represents the elementary lattice T _Q which corresponds to FIG. generator g ₀ = 0110100. Figure 3 shows the elementary mesh 7 which corresponds to the generator g = 1011000.

 Let be two elementary lattices Tj and T, (i ≠ j) defined as:

T, = {(sie ^Î sr ¹ )}.

 Tj = ίΟί Γ)} ·

 The product of these two elementary lattices denoted by T; is defined by:

W ^ KC ^s fsfJ. CefeeO. Cs S sH)} -

The product of two lattices thus performs the Cartesian product of their states and the sum of the transitional branch labels between these states. The product lattice T _Q ^ résultant resulting from the product of the two elementary lattices T ₀ and T _j respectively shown in FIGS. 2 and 3 is represented by FIG. 4. FIG. 4 represents the product _Q of the two T ₀ lattices. and previous TL in the case of a generator matrix.

 Elementary lattices can just as easily be constructed from the code parity check matrix. A control matrix H of a block code C (n, k) is a matrix of size (n-k) x n. H is defined such that for every codeword c of C, it checks:

 0 (mod 2).

 Each line of H is associated with an elementary lattice with two states 7j. The global lattice T representing the code C is obtained by the Cartesian product of all (n-k) elementary lattices:

In the case of a control matrix H of the code, the construction of an elementary lattice and the product of two lattices are described from the example of the Hamming code n = 8, k = 4, d-min- 4) whose control matrix ¾ _x8 is given by:

An elementary lattice Tj represents the ith line of the control matrix H of the code. Each elementary lattice is constituted by a sequence of sections "section 0" and "section 1" according to the bit values of a line or generator of the matrix H defined by FIGS. 5a and 5b. FIGS. 5a and 5b respectively represent the sections associated with the two bit values (bit = 0 and bit = 1) of a line of the control matrix H of the code. For the example of the preceding Hamming code (8.4.4), the two elementary lattices associated with the first two rows of its control matrix are represented respectively by FIGS. 6a and 6b. FIG. 6a represents the elementary lattice T ₀ corresponding to the generator 10000111. FIG. 6b represents the elementary lattice T corresponding to the generator 01001011.

The product of two elementary lattices associated with two lines of a control matrix is defined below. Let be two elementary lattices Tj and T _j (i ≠ j) defined such that:

T _i = {(si, ef _jS )} and Ti = {(sf, ef, s)}.

The product of these two elementary lattices of a control matrix denoted Tj®T _j is defined by:

Τ, βΓΓ, = {((sf, sj), (ef, e [), ef.

The product of two lattices thus carries out the Cartesian product of each of the three components of the triplet-branches but by suppressing the branches if e [≠ ef. The product lattice T ₀ ®T ₁ resulting from the product of the two elementary lattices T ₀ and Tj represented respectively in FIGS. 6a and 6b is represented by FIG. 7.

 The first method of rigorously optimal decoding known is to find exhaustively among all the words c, belonging to the code C, the one that is closest to the received word. This method outputs discrete values of symbols after decision, 0 or 1 if the alphabet is binary, it is said to decision "hard" (hard). This exhaustive decoding method is of exponential complexity in 2 ^, where k is the dimension of the code, and therefore very expensive and difficult to achieve. And so it does not currently allow decoding in real time codes larger than about k> 24.

Among the methods for decoding linear correcting codes using soft information, the Viterbi algorithm is known [3]. This algorithm is used for the real-time decoding of convolutional codes and more generally for the decoding of all the codes described. by a trellis. The Viterbi algorithm consists of looking for the shortest path in a tree built from the encoder lattice. This path therefore corresponds to the most probable sequence that is to say which is at the minimum distance of the received sequence. This received sequence which corresponds to the received word to be decoded, is for example a sequence of n binary data (hard decision). The received sequence may be different from the coded sequence coming from the coder taking into account errors introduced during transmission between the coder used on transmission and the decoder used. at the reception.

The tree used for decoding in the case of a convolutional encoder has a "vertical" dimension equal to the number of possible states of the encoder, ie 2 ^m , where m is the code constraint length, that is to say the number of memory elements used during the coding, and a "horizontal" dimension called depth. At each new entry of a binary data group of the received sequence, the algorithm builds according to a so-called forward phase a new section of the tree from each arrival node of the preceding section taken. as the starting node and to the arrival nodes of the new section. For each current node taken from the starting nodes of the new section of the tree, the algorithm examines all possible branches of the lattice and for each branch it determines a transition metric which is the distance between the branch label and the group of binary data received by the decoder. At each arrival node, if there are several branches that arrive there, the algorithm keeps only the branch that gives the minimum distance; the algorithm systematically eliminates one of two possible paths reaching each arrival node. The algorithm assigns to the arrival node, at the end of the path consisting of a succession of branches, a cumulative metric equal to the sum of the preceding cumulative metric and the transition metric of the selected branch. When the sequence of n binary data received has been fully traversed, the algorithm retains only the arrival node to which the lowest cumulative metric is allocated. Going up the tree from this node during a rising phase called "survivors", the algorithm determines the best path by performing a backward reading of decisions taken. The basic principle of the Viterbi algorithm is in fact to consider in each node of the tree only the most probable path so as to make it possible to easily trace the lattice and thus to determine a posteriori an estimate of the value received. several moments of reception before.

 This Viterbi algorithm uses as input hard information (ie symbols) or soft information (ie the sampled values of the received signal) but only outputs hard information (ie symbols): it is said that it is a "Soft-In-Hard-Out" decoding algorithm.

In the case of a non-recursive binary convolutional encoder, the trellis is formed of nodes connected by branches. The nodes represent the different possible states of the encoder; there are 2 ^m states if the state vector, i.e. the length of the encoder memory, is of dimension m. The branches represent the different possible transitions from one node to another (i.e. from a state of the encoder to the next state of the encoder) upon the arrival of an input bit. From each node, it leaves two branches respectively associated with the arrival of a "0" or a "1" at the input of the encoder.

The encoder state at time t is represented by the vector sj ^ state, ..., S _j ^{771- l '').} At each arrival of a bit x _t at the input of the coder, the coder generates a codeword at the output

(c, c, ..., Cf ^771-1 '' the label assigned to the branch consists of the input binary element x _t and of the code word (°, c, ..., c ^{1 J)} . Just after the encoder goes to the next state represented by the state vector (s t + l ' ^S t + l>'"' ^S Fig. 8 shows an example of a convolutional encoder with m equal to 2. Fig. 9 shows a lattice section of the convolutional encoder of Fig. 8. The lattice of a convolutional encoder having 2 ^m states, according to FIG. For example, the trellis has four states considering that m equals two, and the state machine is invariant in time all sections are identical.The states are indicated for the initial state and the final state, the values of the inputs x _t and coded outputs (c °, c) are noted in labels on the branches transiting between two states.

 Thus, each code word is represented by a different path in the trellis. The composition of the code word is the concatenation of the labels of the branches that constitute the path found, that is to say the shortest path. FIG. 10 represents a lattice formed by abutting three identical sections to the section shown in FIG. 9. The succession of bold lines represents an example of a path (100, 010, 001), that is to say that for the input sequence (100) the obtained codeword is (001001) starting from the initial state (10) and arriving at the final state (00).

 A variant of this Viterbi algorithm was published in 1974, it is called "BCJR" [4]

(Acronym of the names of its inventors: Bahl, Cocke, Jelinek and Raviv). This variant uses as input soft information (probabilities a priori) but outputs soft information or posterior probabilities for each of the bits: it is said that it is a decoding algorithm "Soft-In-Soft-Out" or " SISO ". In this case, there are two phases of trellis routing: a "forward" phase (respectively a "backward" phase) which accumulate the probabilities of the states in the direction of increasing indices t = 0, l, ..., n -1 sections (respectively in the direction of decreasing indices t = nl, n-2, 1.0) and a final phase that merges the forward and backward phase state probabilities with the prior probability of each bit to compute the posterior probability of each bit.

The Viterbi and BCJR algorithms are too complex due to the high complexity of the lattices and in particular their too large number of states, even for codes with modest error correction capabilities. For example, a code (n = 96, k = 48, dmin = 16) would have a minimum tail-biting lattice of 2 ¹² = 4096 states. In the BCJR algorithm the forward state probabilities are noted at _t (m) and computed as below, and the branched prior probabilities Yt (n, m) for a transition of the state m to the state m 'in the section of index t: a _t (m) = Σ _m £ r _t _ ₁ (m') y _t: (m _J m '). Similarly, the "backward" state probabilities are noted t (m) and computed such that: = Σ _m / _{t + 1} (m ') 7 _i (7 _i m').

Finally, the posterior probabilities of the bits are calculated such that:

Pr _ap p (bit _t = v) = Σ _{(in m} - | _{i) £ = = v)} - _t (m) r _t (m, m ^' ) _t (m') Among the decoding techniques known to decision soft allowing correction transmission errors, we distinguish that of Claude Berrou and Alain Glavieux of 1991 which is a soft-decision iterative decoding method of "Turbo codes" [5] called SOVA and that associated with the decoding of the LDPC codes. The base of the soft-decision decoding of Turbo codes and LDPC codes is the Probability Propagation Algorithm (BP) [2] invented by Robert Gallager in 1961. This BP algorithm propagates probabilities along the branches. a graph of the code and aggregates them on the nodes of this same graph. This graph associated with the code is called the Tanner graph [6] of the code, it represents all the algebraic constraints that the symbols of a codeword must satisfy.

 Turbocodes and LDPC codes (English Low-Density Parity Check) have very good performance in terms of error correction for codes of great length n, with n of the order of a few thousand bits at least (n> 1000).

 On the other hand, turbo codes and LDPC codes have lower performances for codes of shorter length n (n <1000) called short codes.

 This is partly due to the fact that the BP probability propagation algorithm used in the decoding or its variants becomes suboptimal as the minimum length of the cycles decreases. It is recalled that a cycle corresponds to a closed path in a Tanner graph representing the constraints that the symbols of a word must fulfill to be a word of the code. The notion of Tanner graph is well known, and in particular described in the article [7]. Indeed, this BP algorithm is optimal only on a graph having the topology of a tree or a forest of trees, the presence of short cycles greatly degrades the performance of the soft decision iterative decoding algorithm BP.

Such a probability propagation algorithm is therefore not effective for codes having very dense generator matrices (ie having many bits at 1) and therefore for codes having excellent minimum relative Hamming distances δ = d _{m [ n} I n.

 In addition, for industrial use, optimization of error correcting codes includes minimizing coding and decoding complexities in terms of hardware and energy costs consumed. However, these problems of optimization of the decoding of the error-correcting codes are even more difficult to solve when the code lengths are small to medium, that is to say of the order of n <1000.

 There is therefore a need for a new decoding technique good error correcting codes in particular, that is to say error correcting codes having a minimum distance as large as possible, and in particular having very short cycles.

 Presentation of the invention

The invention provides a soft code decoding method of coded symbols, including source symbols and redundancy symbols. These redundancy symbols are obtained by applying, to the coding, an error correction code (n, k) to the source symbols, setting implement a generator matrix G of size kn or a parity check matrix H of size (n-k) .n.

 According to the invention, the method comprises:

 a multistage calculation phase, using elementary lattices Tj determined from the rows of the matrix, each lattice Tj being described by a structure of n sections constructed from different base cells associated respectively with the different symbols of the alphabet and being described by the set of probabilities of its states comprising per stage a calculation step implementing a butterfly operator:

 - to determine distinct elementary lattice products, to compute on a lattice produces posterior probabilities of the states of the lattice product knowing the probabilities of lattice element lattice states and

 for calculating posterior probabilities of the states of each of the lattice lattice products knowing the posterior probabilities of the lattice states produced,

 a phase of determination of posterior probabilities associated with the decoded symbols from the posterior probabilities of the states of all the elementary lattices of the last stage.

The invention thus proposes a new technique for decoding symbols coded with the aid of an error correction code providing soft decisions a posteriori per bit while having a complexity that increases in a polynomial or quadratic manner because of the function of k ^2. and which therefore does not explode exponentially, that is to say in 2 ^k , with the dimension k of the code contrary to the known methods.

Indeed, the method according to the invention does not make the product of k elementary trellises between them to obtain a lattice product global 2 ^k states as described for example by [1].

According to the embodiment, each elementary lattice corresponds to a row of the matrix or results from the product of at least two lines with the constraint that the calculation phase has at least two stages, so k≥ 8 if an elementary lattice is the product of two lines. In a binary alphabet, an elementary lattice thus has 2 ^m states with m the number of lines participating in the product to obtain the elementary lattice.

According to the invention, the method performs for a given stage products bearing only on at least two distinct elementary lattices. The butterfly operator is a calculation operator which by a posterior probability calculation of the states on the product of at least two lattices ensures an interaction between these distinct elementary lattices. In addition, particularly noticeably, the method does not internally use extrinsic probabilities on the bits of the code words but determines posterior probabilities of states. This fundamental characteristic makes it possible to obtain optimal estimates of bit probabilities only at the end of the algorithm and thus makes it possible to overcome the problem of short cycles in a Tanner graph of the code implemented in the known "Belief-Propagation" type decoding methods as used for Turbo codes and LDPC codes and thus allows to decode in real time codes of short lengths (n <1000).

 The state probability propagation is fundamentally different from the existing bit probability propagation in a conventional Belief Propagation algorithm, and allows easy decoding of codes regardless of their length.

 In addition, the invention makes it possible to reduce the number of iterations implemented during decoding, in particular compared to a "Belief Propagation" algorithm.

 Because of this elementary lattice structure which makes it possible to estimate the posterior probabilities of states of these elementary lattices, the recursive estimation of the state probabilities of these elementary lattices is insensitive to cycles in the network of operators. -butterfly.

 According to one embodiment, the calculation step implements at least one iteration of calculation of the posterior probabilities of the states of the lattice product knowing the probabilities of the states of the elementary lattices of the product and calculation of the posterior probabilities of the states of each one. lattices of the product knowing the posterior probabilities of the lattice states produced.

According to this embodiment, the step of calculating posterior probabilities proceeds iteratively. Thus, the method can perform a first calculation of the posterior probabilities and can iterate one or more times this calculation. According to a simple embodiment, the probabilities of the received samples are taken as a priori probabilities are taken into account as initialization values in the calculations of the ^1st floor of the first iteration. According to another embodiment, the probabilities of the samples received taken as prior probabilities are injected in whole or in part in the other stages. The simulation results highlight an iteration gain for certain implementation conditions. More specifically, taking into account an erase channel that typically models packet losses for Internet applications and considering a code with a minimum distance dmin, the theoretical maximum number of erasures is (dmin-1). This maximum correction capacity is obtained with two iterations of the posterior probability calculations.

According to one embodiment, the decoding method further comprises an initialization phase during which for the ^1st stage the probabilities of the states of the elementary lattices are initialized with the prior probabilities of the received symbols and for the following stages the state probabilities of elemental lattices are initialized to ½.

These initialization values make it possible to obtain good estimates by using only the values y received once. The algorithm then recursively performs these computations on the state probabilities of the k lattice, and performs virtual propagation and averaging. of these state probabilities on the global lattice of all elementary lattices.

 According to one embodiment, the calculation of the posterior probabilities of the lattice states produced comprises:

 a propagation phase before, consisting in propagating state probabilities along the branches of the lattice produced and

 a rear propagation phase, consisting in propagating state probabilities along the branches of the lattice produced.

 In other words, the calculation of the posterior probabilities of the states of the elementary lattice product includes a first propagation phase before state probabilities in all the elementary trees, and a second phase of return propagation of the state probabilities in all the elementary trees, each phase being able to proceed in parallel, the process finally combines the probabilities of state before and back for the same section of the lattice produced and for the same state.

 According to one embodiment, the calculation of the posterior probabilities of the states on the lattice product is carried out according to a BCJR or Viterbi-SOVA type algorithm.

 According to one embodiment, the calculation of the posterior probabilities of the states of each of the elementary lattices of the lattice product is performed by marginalization from the posterior probabilities of the lattice states produced. Marginalization is a technique known to those skilled in the art [8].

 According to one embodiment, an elementary lattice corresponds to at least one line of the generator matrix or of the parity check matrix.

 According to one embodiment, the trellis products intervene on sets of at least two elementary trellises. The number of stages and the sets of at least two elementary lattices involved in the lattice products are determined by applying the rules below:

 per stage, the set of lattice indices of sets of at least two elementary lattices form a permutation of the set {0, 1, · · -, k-1],

 any combination of at least two elementary lattices only appears once in all the calculation stages,

- there is at least one path between any operator butterfly ^1st floor and all butterfly operator of the top floor.

 The existence of the path ensures that any elementary lattice influences any other elementary lattice. The set of previous rules allows to interfere a minimum number of times the elementary lattice two by two (directly or indirectly) but sufficient to obtain a good final estimate.

According to one embodiment, the trellis products intervene on trellis pairs. elementary and the lattice pairs involved in the lattice products are determined for the ^1st stage:

 by performing a partition in pairs starting from a permutation identity of order k which makes it possible to obtain a matrix of k / 2 rows and two columns,

 by transposing the matrix obtained to obtain a matrix with two lines and k / 2 columns and flattening this matrix to obtain a list of pairs, and for a following stage:

 repeating the previous operations starting from the list of couples of the previous stage to obtain successive permutations.

 According to this last embodiment, if k is not a power of two, the last permutation is calculated by considering an affine permutation: i-> 5 * i + 1 (mod k).

 According to one embodiment, the alphabet is binary.

 According to one embodiment the number of stages is the rounded up value of Log (2, k).

According to one embodiment the basic cell cello associated with one bit of the same matrix to zero is shown in Figure 5a and the basic unit _cell} associated with one bit of the same matrix a is shown in Figure 5b.

According to the latter embodiment, the product of elementary mesh is determined from the product of the basic cells: cell ® _{0 0} cell, cell _{0 v} ® cell, cel ® cell ₀ and cell cel _®} res ectively equal to:

 A decoding method according to the invention can be used to decode in particular a Hamming code (8.4.4). The number of stages is two and the set of lattice indices of the lattice pairs form the following permutations:

A decoding method according to the invention can be used to decode in particular a Golay code (24, 12, 8). The number of stages is four and the set of lattice indices of the lattice pairs form the following permutations:

 A decoding method according to the invention can be used to decode in particular a QR code (48,24,12). The number of stages is five and the set of lattice indices of the lattice pairs form the following permutations:

LL''iinnvveennttiioonn iinn aa oouuttrree ffoorr oobbjjeett AANN ddééccooddeeuurr dd''uunnee ssééqquueennccee dd''éécchhaannttiilllloonnss rreeççuuss yy ccoorrrreessppoonnddaanntt aapprrèèss ttrraannssmmiissssiioonn ,, ,, ttoo tthhee uunnee ssééqquueennccee cc ccooddééee ssyymmbboolleess CC _dd yy == ,, 00,, 11 ,,

obtained by applying to the coding a linear error correction code of parameters (n, k) with k source symbols implementing a kn dimension generating matrix or a size parity check matrix (nk). symbols belonging to a given alphabet. The decoder includes:

calculating means adapted to perform a multi-stage calculation, using elementary lattices Tj determined from the rows of the matrix, each lattice T; being described by a structure of n sections constructed from different base cells associated respectively with the different symbols of the alphabet and being described by the set of probabilities of its states comprising by stage a calculation step implementing a butterfly operator, to determine distinct elementary lattice products, to calculate on a lattice produces posterior probabilities of the states of the lattice product knowing the probabilities states of the elementary lattices and for calculating posterior probabilities of the states of each of the lattice members of the lattice product knowing the posterior probabilities of the states of the lattice produced,

 computation means adapted to determine posterior probabilities associated with the decoded symbols from the posterior probabilities of the states of all the elementary lattices of the last stage.

 Such a decoding device is particularly suitable for implementing the decoding method described above.

 Such a decoder can of course include the various characteristics relating to the decoding method according to the invention, which can be combined or taken in isolation. Thus, the characteristics and advantages of this decoder are the same as those of the decoding method, and are therefore not detailed further.

 It is noted that such a decoder can be implemented in the form of a digital or analog integrated circuit, or in an electronic component of the microprocessor type. Thus, the decoding algorithm according to the invention can be implemented in various ways, in particular in hard-wired form or in software form.

 In particular, it is possible to use a decoder as described above either to decode coded symbols or to code source symbols.

 In the latter case, the prior probabilities of the received symbols are initialized with the values of the symbols to be coded in the case of binary symbols and the redundancy symbols are forced with a probability of 1/2 in order to be able to use the decoder as an encoder. . In the case of symbols belonging to a non-binary alphabet, the non-binary symbols are considered as vectors. For example Z4 = {0,1, 2,3} is denoted {{0,0}, {0,1}, {1,1}, {1,0}} and the probability of a non-binary symbol is then a vector of four probabilities {ρθθ, ρθΐ, ρΐ ΐ, ρΐθ} whose sum is one. If a symbol is unknown then its vector of probabilities is {1/4, 1/4, 1 / 4,1 / 4}.

 The invention therefore further relates to a method for encoding source symbols, delivering coded symbols comprising the source symbols and redundancy symbols.

According to the invention, such a method for encoding k source symbols delivers a coded sequence of symbols Cj, j = 0, 1, ^■■ ·, (¾-1), obtained by applying a linear error correction code of parameters (n, k) implementing a k-line generator matrix or a parity check matrix at (nk) lines of the coded symbols, the symbols belonging to a given alphabet. The method comprises: a multistage calculation phase, using elementary lattices Tj determined from the rows of the matrix, each lattice Ti being described by a structure of n sections constructed from different base cells associated respectively with the different symbols of the alphabet and being described by the set of probabilities of its states, comprising by stage a calculation step implementing a butterfly operator:

 to determine distinct elementary lattice products, to compute on a lattice produces posterior probabilities of the lattice product states knowing the probabilities of lattice element lattice states and

 for calculating posterior probabilities of the states of each of the lattice lattice products knowing the posterior probabilities of the lattice states produced,

 a phase of determining posterior probabilities associated with the symbols coded ¾ from the posterior probabilities of the states of all the elementary lattices of the last stage,

an initialization phase during which for the ^1st stage the probabilities of the states of the elementary lattices are initialized with the k source symbols and with nk probabilities equal to ½ representing the symbols of redundancy.

 The invention thus proposes a new source symbol encoding technique.

 It is thus possible to simply obtain new codes or reconstruct existing codes, having any length, easily decodable by the decoding method according to the invention.

 Such a coding method can of course include the various characteristics relating to the decoding method according to the invention.

 The invention further relates to a corresponding encoder. Such an encoder of k source symbols delivers a coded sequence of symbols Cj, J '= 0.1, · · ·, («- l), obtained by applying a linear error correction code of parameters (n, k) implementing a k-line generator matrix or a parity check matrix at (nk) lines of the encoded symbols, the symbols belonging to a given alphabet. The encoder includes:

calculation means adapted to perform a multi-stage calculation, using elementary lattices Tj determined from the rows of the matrix, each lattice Tj being described by a structure of n sections constructed from different base cells associated respectively with the different symbols of the alphabet and being described by the set of probabilities of its states, comprising by stage a calculation step implementing a butterfly operator: to determine distinct elementary lattice products, to compute on a lattice produces posterior probabilities of the states of lattice product knowing the probabilities of lattice element lattice states and

 for calculating posterior probabilities of the states of each of the lattice lattice products knowing the posterior probabilities of the lattice states produced,

 calculation means adapted to determine posterior probabilities associated with. coded symbols Cj from the posterior probabilities of the states of all the elementary lattices of the last stage,

calculating means adapted to initialize the probabilities of the elementary lattice of statements ¹ computation stage with the k source symbols and nk probabilities equal to ½ representing redundant symbols.

 Such an encoder is particularly suitable for implementing the coding method described above.

 Such an encoder can of course include the various characteristics relating to the coding method according to the invention, which can be combined or taken in isolation. Thus, the characteristics and advantages of this encoder are the same as those of the coding method, and are therefore not detailed further.

 Again, such an encoder can be implemented as a digital or analog integrated circuit, or in an electronic microprocessor component. Thus, the coding algorithm according to the invention can be implemented in various ways, in particular in hard-wired form or in software form.

 The invention also relates to a device comprising a coder and a decoder of previously described symbols.

 Such a device, also called coded coder / decoder, can be implemented in the form of a digital or analog integrated circuit, or in a microprocessor-type electronic component. In particular, the use of a codec makes it possible to pool the hardware resources used for coding and decoding. For example, such a codec may receive source symbols as input, and output coded symbols, or receive coded symbols as input, and output an estimate of the source symbols.

 In still another embodiment, the invention relates to one or more computer programs comprising instructions for implementing a decoding method or coding method as described above, when the program or programs are executed by a processor. Such programs can be stored on an information medium.

List of Figures Other characteristics and advantages of the invention will emerge more clearly on reading the following description of particular embodiments, given as simple illustrative and non-limiting examples, and the appended drawings, among which:

FIG. 1 represents the elementary lattice T _t corresponding to the generator g _t in the case of a generator matrix,

FIG. 2 represents the elementary mesh T ₀ which corresponds to the generator g _Q - 0110100 for the example of the Hamming code (7,4,3),

FIG. 3 represents the elementary lattice 7 which corresponds to the generator g _x = 1011000 for the example of the Hamming code (7,4,3),

FIG. 4 represents the product T _Q of the two preceding lattices ₇₀ and 7 in the case of a generator matrix,

 FIGS. 5a and 5b respectively represent the sections associated with the two bit values (bit = 0 and bit = 1) of a line of the control matrix H of the code,

FIG. 6a represents the elementary mesh T ₀ corresponding to the generator 10000111 for the example of the Hamming code (8.4.4),

FIG. 6b represents the elementary mesh T _x corresponding to the generator 01001011 for the example of the Hamming code (8.4.4),

FIG. 7 represents the product lattice T _Q ® 2 resulting from the product of the two elementary lattices T ₀ and T _x represented respectively in FIGS. 6a and 6b,

 FIG. 8 is a diagram of an example of a convolutional encoder with a memory length m equal to two,

 FIG. 9 represents a trellis section of the convolutional coder of FIG. 8, FIG. 10 represents an example of a trellis with three four-state sections of the convolutional coder of FIG. 8,

FIG. 11 diagrammatically represents a coder COD and a decoder DECOD, FIGS. 12a and 12b represent the two elementary lattices T ₀ and ΤΊ respectively corresponding to the first two lines of a generator matrix G of the Hamming code (n = 8, k = 4, d _min = 4),

 FIG. 13 represents the main steps implemented by an encoder / decoder according to the invention,

 FIG. 14 represents the lattice product for the example of the Hamming code (n = 8, k = 4; FIG. 15 represents an operator-butterfly network for an encoder / decoder of a code defined by a matrix comprising 4 lines according to the invention,

FIG. 16 represents the diagram of a complete coder / decoder according to the invention for a code defined by a matrix comprising 4 lines according to the invention (example the Hamming code (8.4.4), FIG. 17 represents the diagram of a complete coder / decoder according to the invention for a code defined by a matrix comprising 12 lines according to the invention (example the Golay code (24, 12, 8),

 FIG. 18 represents the diagram of a complete coder / decoder according to the invention for a code defined by a matrix comprising 24 lines according to the invention (example the QR code (48, 24, 12),

 FIGS. 19a and 19b are diagrams of the simplified structures of an encoder and a decoder according to one embodiment of the invention, in particular for the case of elementary two-state lattices and butterfly operators processing sets of 2 ( ie couples) elementary lattices.

 Description of Embodiments of the Invention

The invention performs for a given calculation stage products relating only to sets of at least two distinct elementary lattices and does not perform multiplication of k (respectively (nk)) elementary lattice between them to obtain a lattice global product 2 ^k (respectively 2 ^{(n "!;))} states. the invention makes interact elementary mesh each other by means of a throttle operator. the propagation of interactions is made of extremely advantageously on state probabilities and not on bit probabilities which makes it insensitive to the cycles of the structure of the operator-butterfly network.

 By way of example, the invention offers a decoding technique that offers better performance than the techniques of the prior art with respect to good error-correcting codes such as: conventional algebraic codes, such as the Golay code (24,12,8), the QR code (48,24,12); turbo-codes; LDPC codes, etc.

 The invention codes or decodes symbols belonging to a given alphabet. This one can be the binary alphabet. The following description is based on a binary alphabet knowing that the alphabet may just as well not be binary.

 The main steps implemented by a coder and / or a binary decoder according to the invention are described in relation with FIG. 11.

 This is a simple example, in which the invention is used for both encoding and decoding symbols. Of course, it is possible to use the proposed decoding algorithm with a classical linear error correction coding algorithm of parameters (n, k) implementing a generator matrix G or of n-column H parity control and respectively k lines or (nk) lines of symbols.

Thus, according to the embodiment illustrated in FIG. 11, a coder COD according to the invention takes as input k source data x and outputs a coded sequence of data Cj, j-0.1, · · ·, (" - l) comprising the source data x and redundancy data r. The linear error correction code of parameters (n, k) implements a matrix of binary data, this matrix being either generatrix G of dimension kxn or of parity check H of dimension (nk) xn.

If the coding is systematic, the generating matrix G is then said in systematic form, the code word c comprises the source bits x = (x ₀ , x _lt ..., χ ^)) corresponding to the useful information placed generally on the left and the redundancy bits r = (r ₀ , r _ïr ..., r ( _n _ _{k - 1} ) placed in a row or vice versa: c = (x ₀ , x _lr ..., a ^ -i), r ₀ , r _x , ..., r ( _n _ _{¾ - 1} )). A code word c is such that: c = xG or cH ^tr = 0.

 The coded sequence c may be conveyed in a transmission channel CH or stored on a medium, and then decoded by a decoder DEC according to the invention. Such a decoder receives as input a sequence y of samples comprising the source data x and redundancy data r possibly tainted with error and delivers an estimate x of the source data.

 According to one embodiment of the elementary lattices, each elementary generator or line of the matrix G respectively H is associated with a two-state elementary lattice, given that the symbols are taken from the binary alphabet {0, 1} with two symbols. To generalize to any type of alphabet, each elementary lattice thus has as many states that the alphabet contains symbols.

 The invention preserves the k (respectively (n-k)) elementary lattices Ti with at least two states corresponding to the k (respectively (n-k)) lines L; of the matrix G (respectively H) and produces products on sets of at least two (i.e. distinct) elementary lattice pairs. Each elementary lattice is described by a structure of n sections of index t, a section consisting of branches each represented by a triplet (starting state, branch label, arrival state) and by the set of probabilities of its components. states: {f} (sf)}.

A method according to the invention uses the k (respectively (nk)) elementary lattice T The structure of the lattice T _; elementary is constructed according to the invention from different basic cells respectively associated with the different symbols of the alphabet ensuring a minimum number of lattice states.

According to one embodiment, the basic cell cello associated with a bit of the matrix equal to zero is as follows:

 <D

and the cell base cell; associated with a bit of the matrix equal to one is as follows:

In cello and ce¾, the states denoted 0 and 1 are the states of entry and exit of a section in the horizontal direction.

 A trellis is formed by the end-to-end sections themselves composed of branches. A branch of a trellis section is a triplet (starting state, branch label, state of arrival). A lattice can therefore be seen as a set of branches:

The invention is illustrated with the example of the Hamming code (n = 8, k = 4, d _min = 4). A generator matrix G of this Hamming code is as follows:

The Hamming code (n = 8, k = 4, d _min = 4), and in general any code can be described completely and unambiguously in an equivalent way by a control matrix H or by a set of equations modulo two, which is the following for the Hamming code:

r ₀ + _Xl + x ₂ + x ₃

Each equation is associated with a two-state elemental lattice. The elementary lattice T ₀ associated with the generator 10000111 which corresponds to the first line L ₀ of the matrix G and therefore to the boolean equation r ₀ + x _] + x ₂ - x ₁ -0 is thus illustrated by FIG. 12a.

The elementary lattice T [associated with the generator 01001011 which corresponds to the second line Li of the matrix G and to the boolean equation + x ₀ + x ₂ + x _i = 0 is thus illustrated by FIG. 12b.

 For the sake of simplicity, k is considered to be the number of rows in the matrix.

 A method 1 according to the invention, the main steps of which are illustrated in FIG. 13, comprises a phase 2 of multistage calculation 3 which uses the elementary k trellises Ti. By stage 3, the method comprises a calculation step 4, 5, 6 which implements a butterfly operator.

During the calculation step, the butterfly operator determines 4 festive products. of distinct elementary lattices, T _t and T _j with i ≠ j. The product 2 ^ ®7j of two lattices of the same number n of sections of index t G {0,1, ..., n-1} is, according to Calderbank-Forney-Vardy [l], such that :

T _j ®T _j - ë _} . So, the product of two

lattice is the Cartesian product of each of the triplet components of the branches taken in pairs with the suppression of pairs of branches such as

e _j '. According to the invention the elementary mesh product is determined from the product of the basic cells: cell ₀ ® cell _0, cell _Q ® cell _x, cell ₀ and cell cel ® ® cell _x respectively equal to:

Taking the example of the Hamming code (n = 8, k = 4, d _min = 4), the product T _Q ®7 is illustrated in FIG.

According to another embodiment of the elementary lattice, considering k the number of rows of the matrix, an elementary lattice is associated with a number m> l of rows of the matrix defining the code, its number of states per section is then 2 ^m . If an elementary trellis Ti is associated with (g0, gl, ..., gm-1) lines of the matrix then it is itself equivalent to the product of the trellis Ti each associated with a line gi, i = 0 ,. .., ml. This does not change the principles of the algorithm but modifies in particular the number of operator-butterflies per floor which is q = ^^ 2m ^s i 2 xm divides k and decreases the number of stages since the interactions are between lattice more complex elementals associated with more rows of the matrix. If 2 X m does not divide k then k = qx 2 xm + r with 0 <r <2m. Each stage comprises q operator-butterflies interacting two elementary lattices at 2 ^m states each associated with m rows of the matrix and a butterfly operator. If r = m + rr, this last butterfly operator makes an elementary lattice interact with 2 ^m states and an elementary lattice with 2 ^π states associated with rr lines of the matrix, if r <m this last operator-butterfly makes interact two trellises associated to the remaining m lines. The most interesting cases apart from the case m = 1 which corresponds to a particular embodiment of the elementary lattices, are the cases m = 2, m = 3 and m = 4 to keep a reasonable computational complexity and good performances of convergence towards the optimal estimation of the posterior probabilities of lattice states.

 During the calculation step, the method makes the lattices interact at least two by two

(directly or indirectly) by means of the butterfly operator a minimum number of times but sufficient to obtain a good final estimate and so that any elementary lattice influences any other elementary lattice. In particular, the interactions are determined such that:

any butterfly operator performs interactions only between sets of at least two (here pairs of) lattices Ti and Tj distinct, i ≠ j, the set of butterfly operators of the same stage covers, in a bijective manner, all the lattices: ie the indices of the lattices of one stage form a permutation of {0, 1,. . . , k - 1},

 - globally, everything together (in the simplest case: any pair) of elementary lattices appears only once and only once,

 - There is at least one path between any butterfly operator on the top floor and any butterfly operator on the first floor.

The interactions may be in the form of an overall matrix of at least two (in one embodiment: pairs of) elementary lattice indices that interact by means of a butterfly operator. The sets of at least two (according to one embodiment: lattice pairs) involved in the lattice products are determined for the ^1st stage:

 by performing a partition of sets of at least two indices (according to one embodiment: in pairs) from an identity identity permutation k that makes it possible to obtain a matrix of k / 2 rows and two columns ,

 by transposing the matrix obtained to obtain a matrix with two lines and k / 2 columns and by flattening this matrix to obtain a list of sets of at least two indices (according to one embodiment: couples),

 and for a next floor:

 repeating the preceding operations starting from the list of sets of at least two indices (according to one embodiment: couples) of the preceding stage to obtain successive permutations.

 The process therefore determines a set of nEtage = \ Log (l, ki \ permutations of the set of indices {0, 1, ..., k - 1} where the operator \.] Is the rounded up value. .

Taking again the example of the Hamming code (n = 8, k = 4, d _min = 4), nEtage = 2 since k = 4 and the set of permutations is described by a matrix of two lines and two columns of couples of clues:

For the Hamming code (n = 8, k = 4, d _mi - = 4), the calculation phase has two floors, it corresponds to the ^re column and the 2 ^nd and last stage corresponds to the second column. In other words, in the example of the Hamming code (n = 8, k = 4, d _min = 4) the method made to interact at the floor of the ^Γ couples calculation phase _(Q T, and Tj (T _2, T ₃ ) and during the 2 ^nd stage couples (T ₀ , T ₂ ) and TT ₃ ) c & which can be represented by the diagram of Figure 15.

According to the invention, the butterfly operator furthermore performs a calculation of the posterior probabilities of the states on the product ®T _j of two lattices T _{ and 7 \ with i ≠ j, then a calculation 6 of the posterior probabilities of the states of each of the two lattices T _t and T-.

In one embodiment, Γ operator butterfly calculates the posterior probabilities of the states on the product of the two trellis T _t - and _T} using a BCJR algorithm [4] adapted to obtain as output result vector the lattice state probabilities produced while the classical BCJR [4] algorithm provided the vector of the posterior probabilities of the bits.

According to one embodiment, the throttle operator performs the calculation of the posterior probabilities of the states of each of the two lattices T _{ and T- by marginalization.

According to one embodiment, a decoding method according to the invention comprises an initialization phase. According to a simple embodiment of the initialization, the probabilities of the states {Pr (s)} of the elementary lattices T; i = 0, 1, · · ·, (τΐ- k- 1) are initialized only for the ^1st stage with the prior probabilities of the received bits y. For the following stages, the probabilities of the states {r (sf)} of the elementary lattices T; i = 0, 1, · · ·, (η- k- l) are initialized to ½. The values of the received bits y are therefore only injected once in the decoding process. According to one embodiment of the initialization, the values of the received bits are injected in whole or in part as prior probabilities received in the other stages.

Considering that a butterfly operator has in input data the probability vectors of states {Pr (s')} and {Pr (sJ)} of the elementary lattices T _; and T _j with i ≠ j, these vectors correspond to the posterior probability vectors of the states of each of the two lattices T _t and T _j determined in the preceding stage, for each of the stages following the first stage.

 The decoding method according to the invention further comprises a phase of determining posterior probabilities associated with the decoded symbols from the posterior probabilities of the states of the elementary k trellises of the last stage. This determination amounts to merging all the information available on each decoded symbol between the elementary k trellises of the last stage:

 & -1

PRP jy _j = 1) Pr oc JJ (c _j = 1 j ^"XL y)

i = 0 with "the equality operator with normalization (division by the sum of the probabilities of the two cases" 0 "and" 1 ".

According to one embodiment, this merger corresponds to the multiplication of all the final probabilities of one and the same code bit of each of the elementary lattices T _s - with i = 0, 1, - · ·, (η - & - 1) from the posterior probabilities of the states of the elementary k lattice of the final stage.

According to one embodiment, this merger corresponds to the addition of all the final metrics of one and the same code bit of each of the elementary lattices 7 ^ with I = 0.1, - ^■ ·, («- k-1) from the posterior probabilities of the states of the elementary k lattice of the final stage.

So, taking the example of the Hamming code (n = 8, k = 4,

the decoding method can be illustrated by the diagram of Figure 16. The fusion provides the probabilities of the decoded bits: {Pi ~ (Xi \ y)} knowing the received bits y. The structure shown in FIG. 16 is valid for all the codes defined by k = 4 elementary trellises, whatever its length n.

The invention is illustrated with the example of the Golay code (n = 24,

According to the decoding method previously described, the number of stages is nEtage = 4 since k = 12. The set of permutations is described by a matrix of six lines and four columns of pairs of indices:

 The structure of the decoding algorithm is shown in Figure 17. The letters Ai, Bi, Ci and Di refer to the different butterfly operators.

The invention is illustrated with the example of the QR code (n = 48, k = 24, d _min = 12). According to the decoding method previously described, the number of stages is nEtage = 5 since k = 24. The set of permutations is described by a matrix of twelve rows and five columns of pairs of indices:

 The structure of the decoding algorithm is shown in FIG. 18. The letters Ai, Bi, Ci, Di and Ei refer to the various butterfly operators.

 The description of the invention was made by considering probabilities. The invention can just as well consider metrics. In this case, according to one embodiment, a butterfly operator performs the calculation of the posterior states metrics on the product of the two lattices Ti and Tj according to a Viterbi-SOVA type algorithm [5].

 FIGS. 1a and 19b show schematically the simplified structures of a coder COD and a decoder DEC according to one embodiment of the invention.

 The encoder comprises a memory 10 comprising a buffer memory, a processing unit 11, equipped for example with a microprocessor μΡ, and driven by the computer program 12, implementing the coding method according to an embodiment of the invention. 'invention.

At initialization, the code instructions of the computer program 12 are for example loaded into a RAM memory before being executed by the processor of the processing unit 11. The processing unit 11 receives as input source data x and delivers coded data c including the source and redundancy data. The microprocessor of the processing unit 11 implements the steps of the coding method described above, according to the instructions of the computer program 12, for encoding the source data. For this purpose, the encoder comprises, in addition to the buffer memory 10, a coding module for determining products T _j ®Tj of distinct elementary lattices, T _t and Γ- with i ≠ j, to calculate by trellis product T _t ®T _j of the posterior probabilities of the states of the product ® T _j knowing the probabilities of the states ÎPÎ ^* S ')} of the elementary lattices T ₍ and T _j and to compute posterior probabilities of the states of each of the two lattices T _t and T _j knowing the posterior probabilities of the states of the lattice produced and to determine posterior probabilities associated with the coded symbols C. This module is controlled by the microprocessor of the processing unit 11.

 The decoder DEC comprises a memory 20 comprising a buffer memory, a processing unit 21, equipped for example with a μocess microprocessor, and controlled by the computer program 22, implementing the decoding method according to an embodiment of FIG. the invention.

At initialization, the code instructions of the computer program 22 are for example loaded into a RAM memory before being executed by the processor of the processing unit 21. The processing unit 21 receives as input data encoded including source data and redundancy data, and provides an estimate of the source data. The microprocessor of the processing unit 21 implements the steps of the decoding method described above, according to the instructions of the computer program 22, for decoding the coded data. To do this, the decoder comprises, in addition to buffer memory 20, a decoding module for determining ® T _{j j} products distinct elementary mesh, _T; and Γ- with i ≠ j, to calculate by product lattice T _j ®T _j posterior probabilities of the states of the product ®T _j knowing the probabilities of the states {Pr (s)} of the elementary lattices T _t and T. and for calculating posterior probabilities of the states of each of the two lattices T _t and T. knowing the posterior probabilities of the states of the lattice product T _t ®T-, and for determining posterior probabilities associated with the decoded symbols. This module is controlled by the microprocessor of the processing unit 21.

References :

 [1] A.R. Calderbank and G.D. Jr. Forney and A. Vardy, "Minimal Tailbiting Trellises: The Golay Code and More", IEEE Transactions on Information Theory, Vol. ΓΓ.45, no.5, pp.1435-1455, July 1999.

[2] R. G. Gallager, "Low-Density Parity-Check Codes," IEEE Transactions on Information Theory, Vol. 21-28, January 1962.

[3] GD Forney, "The Viterbi Algorithm", Proceedings of the IEEE, Vol.61, no.3, pp.268-278, March 1973. [4] LR Bahl and J. Cocke and F. Jelinek and R. Raviv, "Optimal Decoding of Linear Codes for Minimizing Symbol Error Rate," Transactions on Information Theory, Vol. IT-20, pp. 284-287, March 1974.

 [5] C. Berrou, A. Glavieux, and P. Thitimajshima, "Near Shannon Limit Error-Correcting Coding and Decoding: Turbo-Codes", in Proceedings of the IEEE Int. Conf. on Communications (ICC 93), Geneva, Switzerland, pp.1064-1070, May 1993.

[6] N. Wiberg, "Codes and Decoding on General Graphs," Ph.D. Thesis, Link ^' oping University, Sweden 1996.

 [7] R.M. Tanner's "Minimum-distance bounds by graph analysis", "IEEE Transactions on information theory", vol. 47, February 2001.

 [8] Terrence L. Fine, Pearson Education International, ISBN-10: 0130205915, chapter 11.2.3, pp. 324-325, "Probability and Probability of Electrical Engineering."

Claims

Method (1) for decoding a sequence of samples received y by a decoder corresponding, after transmission, to a coded sequence of symbols c _j , j = 0,1, · ^■ -, [n-1), obtained applying to the coding a linear error correcting code to source symbols (x) implementing an n-column matrix and k symbol lines, the symbols belonging to a given alphabet, characterized in that said method comprises:

 a multistage calculation phase, using elementary lattices Tj determined from the k rows of the matrix, each lattice Ί \ being described by a structure of n sections constructed from different base cells associated respectively with the different symbols of the 'alphabet' and being described by the set of probabilities of its states ({Pr (s)}, i = 0, 1, · · ·, (η-k-1)) comprising per stage a calculation step implementing a butterfly operator:

to determine products (T _t ®Tj) of distinct elementary lattices, (T _t , T.

. i ≠ j),

to compute on a lattice produced (T _j ®T _j ) posterior probabilities of the states of the lattice product (7 ®2 ^™ ) knowing the probabilities of the states

(

elementary lattices (T _t , T.) of the lattice product and for calculating posterior probabilities of the states of each of the lattice product lattice lattices (T _j T _j ) knowing the posterior probabilities of the states of the lattice produced (T _j ®T-),

 a phase of determining posterior probabilities associated with the decoded symbols from the posterior probabilities of the states of all the elementary lattices of the last stage.

Method (1) of decoding according to claim 1 wherein the calculation step implements at least one iteration of calculation of the posterior probabilities of the states of the lattice produced (T _j ®Tj) knowing the probabilities of the states ({Pr ( s)}, {Pr (sJ)}) elementary lattices (7 ^, - -) of the product and calculation of the posterior probabilities of the states of each of the lattices (T _t , T _j ) of the product knowing the posterior probabilities states of the lattice produced (®7). Method (1) for decoding according to claim 1, further comprising an initialization phase during which for the ^1st stage the probabilities of the states of the elementary lattices are initialized with the prior probabilities of the symbols received and for the following stages the state probabilities of elemental lattices are initialized to ½.

Method (1) for decoding according to claim 1, wherein the calculation of the posterior probabilities of the states of the lattice produced ( _t ®T _j ) comprises:

 a propagation phase before, consisting in propagating state probabilities along the branches of the lattice produced and

 a rear propagation phase, consisting in propagating state probabilities along the branches of the lattice produced.

Method (1) for decoding according to Claim 1, in which the calculation of the posterior probabilities of the states of each of the elementary lattices of the lattice product (T _s and Γ.) Is carried out by marginalization from the posterior probabilities of the states of the lattice product (T.TM.).

Method (1) of decoding according to claim 1, wherein a basic lattice corresponds to at least one line of the matrix.

Method (1) of decoding according to claim 1 wherein an elementary lattice corresponds to a row of the matrix and wherein the number of stages is the rounded up value of Log (2, k).

The method (1) of decoding according to claim 1 wherein the alphabet is binary.

9, decoding method (1) according to the preceding claim, wherein the base cell cello associated with a bit of the matrix equal to zero is as follows:

and the basic cell cellj associated with a bit of the matrix equal to one is as follows:

10. Method (1) for decoding according to the preceding claim wherein the elementary lattice product is defined from the products of the base cells: cell ₀ ® cell ₀ , cell ₀ ® cell, cell ® cell ₀ and cell _{ ® respectively equal to:

11. Use of a decoding method according to one of claims 1 to 10 combined with claim 8 for decoding a Hamming code (8.4,4) such that the number of stages is two and all lattice indices of the lattice pairs form the following permutations:

12. Decoder of a sequence of received samples corresponding thereto, after transmission, to a coded sequence of symbols c _j , j = 0.1, · ^■ ·, {η-1), obtained by applying to the coding a code linear error corrector for source symbols (x) implementing an n-column matrix and k symbol lines, the symbols belonging to a given alphabet, characterized in that the decoder comprises:

calculation means adapted to perform a multi-stage calculation, using elemental lattices determined from the k rows of the matrix, each lattice T being described by a structure of n sections constructed from different base cells associated respectively with the different symbols of the alphabet and being described by the set of probabilities of its states

comprising, by step, a calculation step using a throttle operator to determine distinct elementary lattice products (T _j TT-) (T with i ≠ y), to compute on a lattice produced (T _j ®T _j ) posterior probabilities of the states of the lattice product T _: ®T _j ) knowing the probabilities of the states

elementary lattice _(T: T _j) and for calculating posterior probabilities of the states of each of the lattice product lattice (T _t , -Γ-) knowing the posterior probabilities of the states of the lattice produced (T _i ®T.),

 computation means adapted to determine posterior probabilities associated with the decoded symbols from the posterior probabilities of the states of all the elementary lattices of the last stage.

13. A method of encoding k source symbols (x) delivering a coded sequence of symbols Cj, J = 0, 1, · ^■ -, (i-1), obtained by applying a linear error correction code implementing a matrix with n columns and k lines of symbols, the symbols belonging to a given alphabet, characterized in that the method comprises:

a multistage calculation phase, using elementary lattices T determined from the k rows of the matrix, each lattice T; being described by a structure of n sections constructed from different basic cells associated respectively with the different symbols of the alphabet and being described by the set of probabilities of its states ({i (s _t (r _£ ))}) , comprising by stage a calculation step implementing a butterfly operator:

to determine products (T, - ®T _j ) of distinct elementary lattices (T _it T-,

- to calculate a product lattice (T ® T _{j j)} of the posterior probabilities of mesh product states (T _f ® T _j) knowing the probabilities of the states ({Pr (s _{t (£))})} trellises elementary (7T, T) of the lattice product and for calculating posterior probabilities of the states of each of the lattice product lattice (T Γ-) knowing the posterior probabilities of the states of the lattice produced (T _j ®T _j )

 a phase of determining posterior probabilities associated with the coded symbols Cj from the posterior probabilities of the states of all the elementary lattices of the last stage,

an initialization phase during which for the ^1st stage the probabilities of the states of the elementary lattices are initialized with the k source symbols and with nk probabilities equal to ½ representing the symbols of redundancy.

14. Encoder of k source symbols (x) delivering an encoded sequence c of symbols c _j , j- 0.1, · · ·, («- 1), obtained by applying a linear error correction code implementing an n-column matrix and k symbol lines, the symbols belonging to a given alphabet, characterized in what the coder understands:

calculating means adapted to perform a multi-stage calculation, using elementary lattices T, determined from the k rows of the matrix, each lattice T _; being described by a structure of n sections constructed from different basic cells respectively associated with the different symbols of the alphabet and being described by the set of probabilities of its states ({Pr (s _t (Ti))}), comprising per stage a calculation step implementing a butterfly operator:

to determine products (T _t - ®T _j ) of distinct elementary lattices (! 7J, T-, i ≠ j),

to calculate a product lattice (T _j ®Tj) posterior probabilities lattices Product states _{T} ® T _j) knowing the probabilities of the states

elementary lattices {T _; , T _j ) of the lattice product and for calculating posterior probabilities of the states of each of the lattice lattice products (7 ^, Γ-) knowing the posterior probabilities of the states of the lattice product (7j),

 computation means adapted to determine posterior probabilities associated with the coded symbols q from the posterior probabilities of the states of all the elementary lattices of the last stage,

calculating means adapted to initialize the probabilities of the elementary lattice of statements ¹ computation stage with the k source symbols and nk probabilities equal to ½ representing redundant symbols.