Genetic algorithms for finding the weight enumerator of binary linear block codes

Abstract

In this paper we present a new method for finding the weight enumerator of binary linear block codes by using genetic algorithms. This method consists in finding the binary weight enumerator of the code and its dual and to create from the famous MacWilliams identity a linear system (S) of integer variables for which we add all known information obtained from the structure of the code. The knowledge of some subgroups of the automorphism group, under which the code remains invariant, permits to give powerful restrictions on the solutions of (S) and to approximate the weight enumerator. By applying this method and by using the stability of the Extended Quadratic Residue codes (ERQ) by the Projective Special Linear group PSL 2 , we find a list of all possible values of the weight enumerators for the two ERQ codes of lengths 192 and 200. We also made a good approximation of the true value for these two enumerators.

New Efficient and Fast Method to Compute the Weights Enumerators of Some Large Doubly Even Self Dual Quadratic Residue Codes

“…The first main advantage is that without hash techniques, the latest step in the old method [7] is very complex and it is very difficult to find the complete list Lw of codewords of a given weight w because it was necessary to verify if a new codeword isn't in Lw before inserting it in Lw when the size of this latest increases. For solving this problem, the authors of [7] have proposed to use a statistical Monte Carlo method to estimate the value of the size Aw of Lw. This statistical way allows finding an estimation of Aw when it is relatively small.…”

“…The stop criteria of the algorithm A3 isn't deterministic but when this algorithm seems converges to true value, the proposed check method permits to verify if the convergence is reached or not yet. For example, as it is explained in [7], for the QR(191) there exists an integer  such that 31=3040*+209.…”

“…In [5] we have presented the method PWEH(Partial Weights Enumerator with Hash techniques) for finding an approximation of Partial Weights Enumerator by integration of Hash techniques in the PWE(Partial Weights Enumerator) [6] in order to reduce its temporal complexity. In [7], authors have proposed a method to find only an approximation of the weights enumerators of quadratic residue codes especially for the lengths 191 and 199. In [8], the authors proposed the use of the complete weights enumerator in order to deduce the weights enumerator for linear code.…”

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

“…The polynomial (5) is called the partial weight enumerator of radius p of the code C having the weight enumerator A, where p is a positive integer less than n-d [8].…”

“…In [8], the authors used genetic algorithms combined with a Monte Carlo method to find the weights enumerator for some residue quadratic codes. 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).…”