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

Abstract

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

New efficient techniques to catch lowest weights in large Quadratic Residue codes

“…Instead the turbo decoder used in [25], the MIM method (Multiple Impulse Method) [23] uses the OSD decoder of order 3 and injects errors in many positions. This method has permits to find good results in terms of time and precision.…”

“…They concluded that genetic algorithms with a large population size significantly outperformed hill-climbing, tabu search and hybrid techniques (GA -Hill climbing and GA -Tabu Search). In [23] the authors had improve some parameter of GA and get best result for BCH code compared to wallis and simulated Annealing [24] and applied this GA to QR codes of length up to 223.…”

A Fast Method to Estimate Partial Weights Enumerators by Hash Techniques and Automorphism Group

“…In Section 3, we present the proposed method PWEHA. In Section 4, we validate the method PWEHA, we compare it with PWE, we give their results for the BCH (255,191,17), BCH (255,187,19), BCH (255, 179, 21) and BCH (255, 171, 23) codes of unknown weights enumerators and we plot their corresponding analytical performances. Finally, a conclusion and a possible future direction of this research are outlined in Section 5.…”

“…In a second work [15], the authors have combined the Monte Carlo method with the multiple error impulse (MIM) technique [16], [17] to find the partial weights enumerator (PWE) for some binary linear codes, in consequences they obtained an upper bound of error probability for MLD decoder for a shortened BCH codes: BCH (130, 66), BCH (103, 47) and BCH (111, 55).…”

“…The main idea in [15] is to look for a list L w of codewords of the same weight w by using the error impulse method [16], [17] with the ordered statistic decoder [20] as given in the algorithm A1. The list L w is used to approximate the value A w by a Monte Carlo method as given in the algorithm A2.…”