Decoding Of Linear Codes Research Articles

Learning to Decode Protograph LDPC Codes

The recent development of deep learning methods provides a new approach to optimize the belief propagation (BP) decoding of linear codes. However, the limitation of existing works is that the scale of neural networks increases rapidly with the codelength, thus they can only support short to moderate codelengths. From the point view of practicality, we propose a high-performance neural min-sum (MS) decoding method that makes full use of the lifting structure of protograph low-density parity-check (LDPC) codes. By this means, the size of the parameter array of each layer in the neural decoder only equals the number of edge-types for arbitrary codelengths. In particular, for protograph LDPC codes, the proposed neural MS decoder is constructed in a special way such that identical parameters are shared by a bundle of edges derived from the same edge-type. To reduce the complexity and overcome the vanishing gradient problem in training the proposed neural MS decoder, an iteration-by-iteration (i.e., layer-by-layer in neural networks) greedy training method is proposed. With this, the proposed neural MS decoder tends to be optimized with faster convergence, which is aligned with the early termination mechanism widely used in practice. To further enhance the generalization ability of the proposed neural MS decoder, a codelength/rate compatible training method is proposed, which randomly selects samples from a set of codes lifted from the same base code. As a theoretical performance evaluation tool, a trajectory-based extrinsic information transfer (T-EXIT) chart is developed for various decoders. Both T-EXIT and simulation results show that the optimized MS decoding can provide faster convergence and up to 1dB gain compared with the plain MS decoding and its variants with only slightly increased complexity. In addition, it can even outperform the sum-product algorithm for some short codes.

Adaptive Linear Programming Decoding of Nonbinary Linear Codes Over Prime Fields

In this work, we consider adaptive linear programming (ALP) decoding of linear codes over prime fields, i.e., the finite fields $\mathbb F_{p}$ of size $p$ where $p$ is a prime, when used over a $p$ -ary input memoryless channel. In particular, we provide a general construction of valid inequalities (using no auxiliary variables) for the codeword polytope (or the convex hull) of the so-called constant-weight embedding of a single parity-check (SPC) code over any prime field. The construction is based on sets of vectors, called building block classes , that are assembled to form the left-hand side of an inequality according to several rules. In the case of almost doubly-symmetric valid classes we prove that the resulting inequalities are all facet-defining , while we conjecture this to be true if and only if the class is valid and symmetric . Valid symmetric classes impose certain symmetry conditions on the elements of the vectors from the class, while valid doubly-symmetric classes impose further technical symmetry conditions. For $p=3$ , there is only a single valid symmetric class and we prove that the resulting inequalities together with the so-called simplex constraints give a complete and irredundant description of the codeword polytope of the embedded SPC code. For $p > 5$ , we show that there are additional facets beyond those from the proposed construction. As an example, for $p=7$ , we provide additional inequalities that all define facets of the embedded codeword polytope. The resulting overall set of linear (in)equalities is conjectured to be irredundant and complete. Such sets of linear (in)equalities have not appeared in the literature before, have a strong theoretical interest, and we use them to develop an efficient (relaxed) ALP decoder for general (non-SPC) linear codes over prime fields. The key ingredient is an efficient separation algorithm based on the principle of dynamic programming. Furthermore, we construct a decoder for linear codes over arbitrary fields $\mathbb F_{q}$ with $q =p^{m}$ and $m>1$ by a factor graph representation that reduces to several instances of the case $m=1$ , which results, in general, in a relaxation of the original decoding polytope. Finally, we present an efficient cut-generating algorithm to search for redundant parity-checks to further improve the performance towards maximum-likelihood decoding for short-to-medium block lengths. Numerical experiments confirm that our new decoder is very efficient compared to a static LP decoder for various field sizes, check-node degrees, and block lengths.

Characterization of cyclic codes over {ℬ[X;(1/m)Z0]}m > 1 and efficient encoding decoding algorithm for cyclic codes

ABSTRACTThe aim of the paper is twofold. Firstly, cyclic codes of arbitrary length n over the family of semigroup rings are completely characterized in terms of ideals of the rings , where is finite unitary commutative ring. Then, generator matrix for cyclic codes over is derived from their ideal representation. Secondly, an efficient encoding decoding algorithm is presented for polynomial cyclic code of length n based on cyclic code over . Rigorous analyses are performed to examine the efficiency of proposed scheme. Analyses reveal that the proposed algorithm can simultaneously encode and decode m messages of . Furthermore, the proposed technique is capable of detecting and correcting all burst errors up to length md−m and from transmitted codewords of . The newly developed algorithm is also compared with some of the existing coding techniques. It is evident from comparison that the proposed scheme is better than existing coding techniques in Babu and Zimmermann [Decoding of linear codes over galois rings, IEEE Trans. Inform. Theory, 47 (2001)], Nagpaul and Jain [Topics in Applied Algebra, The Brooks/Cole Series in Advanced Mathematics, 2005], Shah et al. [A method for improving the code rate and error correction capability of a cyclic code, Comput. Appl. Math. 32 (2013), pp. 261–274, doi:10.1007/s40314-013-0010-1] and Shah et al. [A decoding method of an n length binary BCH code through (n+1)n length binary cyclic code, An. Acad. Brasil. Ciênc. 85 (2013), pp. 863–872].

Self-checking circuits and decoding algorithms for binary hamming and BCH codes and Reed-Solomon codes over GF(2 m )

We consider problems of detecting errors in combinational circuits and algorithms for the decoding of linear codes. We show that a totally self-checking combinatorial circuit for the decoding of a binary Hamming [n, k] code can be constructed if and only if n = 2 r ? 1, r = n?k. We introduce the notion of a totally self-checking combinational circuit detecting error clusters of size at most µ; for shortened Hamming [n,k] codes, we construct totally self-checking decoding combinational circuits detecting error clusters of size at most µ, 2 ? µ < n?k. We describe single-error protected and self-checking algorithms: the extended Euclidean algorithm and decoding algorithms for binary BCH codes and Reed-Solomon codes over GF(2 m ).

Low-density generator matrix codes for indoor and markov channels

We propose a modified algorithm for decoding of linear codes with low-density generator matrix (LDGM codes) over finite-state binary Markov channels. In order to avoid error floors, a serial concatenation of two LDGM codes is utilized. The hidden Markov model representing the channel is incorporated into the graph corresponding to the code, and the message passing algorithm is modified accordingly. The proposed scheme clearly outperforms systems in which the channel statistics are not exploited in the decoding process, allowing reliable communication at rates which are above the capacity of a memoryless channel with the same stationary bit error probability as the Markov channel. The proposed technique can be successfully applied for real wireless channels that can be modeled with hidden Markov models, such as indoor channels. In this case, the hidden Markov model representing the wireless channel can be estimated jointly with the decoding process

Suboptimal decoding of linear codes: partition technique

General symmetric channels are introduced, and near-maximum-likelihood decoding in these channels is studied. First, we define a class of suboptimal decoding algorithms based on an incomplete search through the code trellis. It is proved that the decoding error probability of suboptimal decoding is bounded above for any q-ary code of length n and code rate r by twice the error probability of its maximum-likelihood decoding and tends to the latter as n grows. Second, we design a suboptimal trellis-like algorithm, which reduces the known decoding complexity of the order of q/sup n min (r,1-r)/ operations to that of q/sup nr(i-r)/ operations for all cyclic codes and virtually all long linear codes. We also consider the corresponding bounds for concatenated codes. An important corollary is that this suboptimal decoding can provide complexity below the lower bounds on trellis complexity at a negligible expense in terms of decoding error probability.

Application of circulant matrices to the construction and decoding of linear codes

The Fourier transform technique is used to analyze and construct several families of double-circulant codes. The minimum distance of the resulting codes is lower-bounded by 2 square root r and can be decoded easily employing the standard BCH decoding algorithm or the majority-logic decoder of Reed-Muller codes. A decoding procedure for Reed-Solomon codes is presented, based on a representation of the parity-check matrix by circulant blocks. The decoding procedure inherits both the (relatively low) time complexity of the Berlekamp-Massey algorithm and the hardware simplicity characteristic of Blahut's algorithm. The procedure makes use of the encoding circuit together with a reduced version of Blahut's decoder. >

On complete decoding of linear error-correcting codes

It is shown that minimum distance decoding of linear codes is accomplished by generating all codewords corresponding to all information sets utilizing the corresponding received coordinates. This was previously shown for the special case of maximum distance separable codes. It is shown that for codes over moderate or larger sized fields this method is more effective than that of generating all codewords.

Generalized t-designs and majority decoding of linear codes

The main properties of the generalized t -designs introduced by Delsarte (1973) are studied and used in a majority decoding method which slightly differs from Massey's threshold decoding. The paper also contains a number of results concerning the existence of such designs in codes and a list of some codes which can be decoded by our method.

Iterative decoding of linear codes by a modulo-2 linear program

We present a new form of linear programming that minimizes the real sum of zero-one variables under constraints that are linear in the field of binary numbers and thus nonlinear in the field of real numbers. This problem is motivated by coding theory where binary sequences are transmitted over noisy communication links. The decoding process which corrects some of the errors caused by noise requires the solution of the stated problem. Simulation results are also presented and they show that the algorithms converge to the correct solution.

Generalized threshold decoding of linear codes

It is proved that every linear code of dimension <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> can be decoded by a threshold decoding circuit that is guaranteed to correct <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</tex> errors if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e \leq (d - 1)/2</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</tex> is the minimum distance of the code. Moreover it is demonstrated that the number of levels of threshold logic is less than or equal to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> by giving an algorithm for generating the decoding logic employing <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> levels.