New Efficient and Fast Method to Compute the Weights Enumerators of Some Large Doubly Even Self Dual Quadratic Residue Codes
Saïd Nouh, Moulay Seddiq El Kasmi Alaoui, Mostafa Belkasmi, Abdelaziz Marzak · International journal of intelligent engineering and systems · 2018
Quadratic Residue codes are among the best codes.They have high capacity of error correction but they are very difficult to enumerate and therefore to analyse.Despite all developed methods in this domain, the weights enumerators of Quadratic Residue codes are known only for lengths less than or equal to 167.For the lengths 191 and 199 only estimations are available.In this paper, we present a new method based on the Multiple Impulse Method (MIM) and hash techniques to find the weights enumerators of Quadratic Residue codes having lengths in the form 8m-1, for an integer m.The proposed method Hash_MIM_Weights_Enumerators is validated on all Quadratic Residue codes of known weights enumerators; its reduced spatial and temporal complexities yields to new important results.So, the weights enumerators for the lengths 191, 199 and 223 are determined.These three codes are the best binary linear block codes in terms of minimum distance known until today and their analytical performances are remained unknowns in more than 60 years ago and they are available now.