Search

This letter proposes a novel scheme which consists of a weight-counting algorithm, the combinatorial designs of the Assmus-Mattson theorem, and the weight polynomial of Gleason's theorem to determine the weight distributions of binary extended quadratic residue codes. As a consequence, the weight distributions of binary (138, 69, 22) and (168, 84, 24) extended quadratic residue codes are given. Index Terms--Quadratic residue codes, t-designs, weight distributions.

Recently, an algebraic decoding algorithm suggested by Truong et al. (2005) for some quadratic residue codes with irreducible generating polynomials has been designed that uses the inverse-free Berlekamp--Massey (BM) algorithm to determine the error-locator polynomial. In this paper, based on the ideas of the algorithm mentioned above, an algebraic decoder for the (89, 45, 17) binary quadratic residue code, the last one not decoded yet of length less than 100, is proposed. It was also verified theoretically for all error patterns within the error-correcting capacity of the code. Moreover, the verification method developed in this paper can be extended for all cyclic codes without checking all error patterns by computer simulations. Index Terms--Error-locator polynomial, inverse-free Berlekamp--Massey (BM) algorithm, quadratic residues codes, unknown syndromes.

The weight distributions of binary quadratic residue codes C can be computed from the weight distribution of a subset of C containing one-fourth (resp., one-eighth) of the codewords in C when the length of the code is congruent to i (resp., -1) modulo 8. An algorithm to determine the weight distributions of binary cyclic codes is given. As a consequence, the weight distributions of (73, 37, 13), (89, 45, 17), and (97, 49, 15) quadratic residue codes are determined precisely. Index Terms--Binary cyclic codes, quadratic residue codes, weight distributions.

A composite number can be factored into either N=mp or N=2^{n} , where p$ is an odd prime and m$ , n\geq 2$ are integers. This paper proposes a method for constructing degree-3 and degree-4 perfect Gaussian integer sequences (PGISs) of an arbitrary composite length utilizing an upsampling technique and the base sequence concept proposed by Hu, Wang, and Li. In constructing the PGISs, the degree of the sequence is defined as the number of distinct nonzero elements within one period of the sequence. This paper commences by constructing degree-3 PGISs of odd prime length, followed by degree-2 PGISs of odd prime length. The proposed method is then extended to the construction of degree-3 and degree-4 PGISs of composite length N=mp$ . Finally, degree-3 and degree-4 PGISs of length N=4 , where $n\geq 3$ . [ABSTRACT FROM PUBLISHER]