Computing the Minimum Distance of Linear Codes by the Error Impulse Method (original) (raw)

Optimizing the free distance of Error-Correcting Variable-Length Codes

2010 IEEE International Workshop on Multimedia Signal Processing, 2010

This paper considers the optimization of Error-Correcting Variable-Length Codes (EC-VLC), which are a class of joint-source channel codes. The aim is to find a prefix-free codebook with the largest possible free distance for a given set of codeword lengths, ℓ = (ℓ1, ℓ2,. .. , ℓM). The proposed approach consists in ordering all possible codebooks associated to ℓ on a tree, and then to apply an efficient branch-and-prune algorithm to find a codebook with maximal free distance. Three methods for building the tree of codebooks are presented and their efficiency is compared.

Unique and Minimum Distance Decoding of Linear Codes with Reduced Complexity

Communications in Computer and Information Science, 2011

 Abstract-We show that for (systematic) linear codes the time complexity of unique decoding is     2/ 2 nRH R O n q  and the time complexity of minimum distance decoding is     2 nRH R O n q  . The proposed algorithm inspects all error patterns in the information set of the received message of weight less than 2 d or d , respectively. Index Terms-nearest neighbor decoding, unique decoding, bounded distance decoding, minimum distance decoding.

On the Computing of the Minimum Distance of Linear Block Codes by Heuristic Methods

The evaluation of the minimum distance of linear block codes remains an open problem in coding theory, and it is not easy to determine its true value by classical methods, for this reason the problem has been solved in the literature with heuristic techniques such as genetic algorithms and local search algorithms. In this paper we propose two approaches to attack the hardness of this problem. The first approach is based on genetic algorithms and it yield to good results comparing to another work based also on genetic algorithms. The second approach is based on a new randomized algorithm which we call "Multiple Impulse Method (MIM)", where the principle is to search codewords locally around the all-zero codeword perturbed by a minimum level of noise, anticipating that the resultant nearest nonzero codewords will most likely contain the minimum Hamming-weight codeword whose Hamming weight is equal to the minimum distance of the linear code.

Error Correction in Linear Codes with Computer

fbe.dumlupinar.edu.tr, 2008

Within the context of this study, we have improved a computer program on the syndrome decoding method for correcting codewords received incorrect. This program generates code from given generator matrix and calculates the hamming distance which appertains to this code, also, it finds Slepian(1960) standart array. It gives a list of the decoding table of code and an error pattern relating to be received incorrect codewords. We have used Maple Computer Algebra for calculations [5]. The Algorithm, we have given, reproduces different results from given generator matrix. In here, we have chosen a test problem whose code word's lengthy is 4. But, it is possible for algorithm that it can make calculations for longer codes.

Error-Correction Capability of Binary Linear Codes

IEEE Transactions on Information Theory, 2005

The monotone structure of correctable and uncorrectable errors given by the complete decoding for a binary linear code is investigated. New bounds on the error-correction capability of linear codes beyond half the minimum distance are presented, both for the best codes and for arbitrary codes under some restrictions on their parameters. It is proved that some known codes of low rate are as good as the best codes in an asymptotic sense.

Minimum-distance bounds for binary linear codes

IEEE Transactions on Information Theory, 1973

This paper presents a table of upper and lower bounds on rl,,,(rr,k), the maximum minimum distance over all binary, linear (n,k) error-correcting codes. The table is obtained by combining the best of the existing bounds on d,,,(n,k) with the mini&n distances of known codes and a variety of code-construction techniques.

Performance comparison of short-length error-correcting codes

2016 Symposium on Communications and Vehicular Technologies (SCVT), 2016

We compare the performance of short-length linear binary codes on the binary erasure channel and the binaryinput Gaussian channel. We use a universal decoder that can decode any linear binary block code: Gaussian-elimination based Maximum-Likelihood decoder on the erasure channel and probabilistic Ordered Statistics Decoder on the Gaussian channel. As such we compare codes and not decoders. The word error rate versus the channel parameter is found for LDPC, Reed-Muller, Polar, and BCH codes at length 256 bits. BCH codes outperform other codes in absence of cyclic redundancy check. Under joint decoding, the concatenation of a cyclic redundancy check makes all codes perform very close to optimal lower bounds.

An algorithm for the computation of the minimum distance of LDPC codes

European Transactions on Telecommunications, 2006

The evaluation of the minimum distance of low-density parity-check (LDPC) codes remains an open problem due to the rather large dimension of the parity check matrix H associated with any practical code. In this article, we propose an effective modification of the error impulse (EI) technique for computation of the minimum distance of the LDPC codes. The EI method is successfully applied to sub-optimum decoding algorithms such as the iterative MAP decoding algorithm for turbo codes. We present novel modifications and extensions of this method to the sub-optimum iterative sum-product algorithm for LDPC codes. The performance of LDPC codes may be limited by pseudo-codewords. There are, however, cases when the LDPC decoder behaves as a maximum-likelihood (ML) decoder. This is specially so for randomly constructed LDPC codes operating at medium to high SNR values. In such cases, estimation of the minimum distance d m using the error-impulse method can be useful to assess asymptotic performance. In short, apart from theoretical interest in achievable d m , the technique is useful in checking whether an LDPC code is poor by virtue of having a low d m. But, if a code has a high d m , simulations would still be needed to assess real performance.

Encoding and decoding for the minimization of message symbol error rates in linear block codes

IEEE Transactions on Information Theory, 1987

Given any fixed linear block code, the error rates for the message symbols depend both on the encoding function and on the decoding map. This research shows how to optimize the choice of a generator matrix and decoding map simultaneously to minimize the error rates for all message symbols. The model used assumes that the distibution of messages is flat and that the distribution of error vectors defining the channel is independent of the message transmitted. In addition, it is shown that, with proper choice of coset leaders, standard array decoding is optimal in this circumstance. The results generalize previously known results on unequal error protection and are sufficiently general to apply when a code is used for error detection only.