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

BCH codes have high error correcting capability which allows classing them as good cyclic error correcting codes. This important characteristic is very useful in communication and data storage systems. Actually after almost 60 years passed from their discovery, their weights enumerators and therefore their analytical performances are known only for the lengths less than or equal to 127 and only for some codes of length as 255. The Partial Weights Enumerator (PWE) algorithm permits to obtain a partial weights enumerators for linear codes, it is based on the Multiple Impulse Method combined with a Monte Carlo Method; its main inconveniece is the relatively long run time. In this paper we present an improvement of PWE by integration of Hash techniques and a part of Automorphism Group (PWEHA) to accelerate it. The chosen approach applies to two levels. The first is to expand the sample which contains codewords of the same weight from a given codeword, this is done by adding a part of the Automorphism Group. The second level is to simplify the search in the sample by the use of hash techniques. PWEHA has allowed us to considerably reduce the run time of the PWE algorithm, for example that of PWEHA is reduced at more than 3900% for the BCH (127,71,19) code. This method is validated and it is used to approximate a partial weights enumerators of some BCH codes of unknown weights enumerators. Keywords—Partial weights enumerator; PWEHA; automorphism group; hash function; hash table; BCH codes