Some Results on the Weight Distributions of the Binary Double-Circulant Codes Based on Primes

This paper presents a more efficient algorithm to count codewords of given weights in self-dual double-circulant and formally self-dual quadratic double-circulant codes over GF(2). A method of deducing the modular congruence of the weight distributions of the binary quadratic double-circulant codes is proposed. This method is based on that proposed by Mykkeltveit, Lam and McEliece, JPL. Tech. Rep., 1972, which was applied to the extended quadratic-residue codes. A useful application of this modular congruence method is to provide independent verification of the weight distributions of the extended quadratic-residue and quadratic double-circulant codes. Using this method in conjunction with the proposed efficient codeword counting algorithm, we are able i) to give the previously unpublished weight distributions of the [76, 38,12] and [124, 62, 20] binary quadratic double-circulant codes; ii) to provide corrections to the published results on the weight distributions of the binary extended quadratic-residue code of prime 151, and the number of codewords of weights 30 and 32 of the binary extended quadratic-residue code of prime 137; and iii) to prove that the [168, 84, 24] extended quadratic-residue and quadratic double-circulant codes are inequivalent