Cyclic codes form an important class of codes. They have very interesting algebraic structure. Furthermore, they are equivalent to many important codes, such as binary Hamming codes, Golay codes and BCH codes. Minimal...Cyclic codes form an important class of codes. They have very interesting algebraic structure. Furthermore, they are equivalent to many important codes, such as binary Hamming codes, Golay codes and BCH codes. Minimal codewords in linear codes are widely used in constructing decoding algorithms and studying linear secret sharing scheme. In this paper, we show that in the binary cyclic code all of the codewords are minimal, except 0 and 1. Then, we obtain a result about the number of minimal codewords in the binary cyclic codes.展开更多
In this paper, we make some progress towards a well-known conjecture on the minimum weights of binary cyclic codes with two primitive nonzeros. We also determine the Walsh spectrum of Tr(x d ) over F2m in the case whe...In this paper, we make some progress towards a well-known conjecture on the minimum weights of binary cyclic codes with two primitive nonzeros. We also determine the Walsh spectrum of Tr(x d ) over F2m in the case where m = 2t, d = 3 + 2t+1 and gcd(d, 2m-1) = 1.展开更多
文摘Cyclic codes form an important class of codes. They have very interesting algebraic structure. Furthermore, they are equivalent to many important codes, such as binary Hamming codes, Golay codes and BCH codes. Minimal codewords in linear codes are widely used in constructing decoding algorithms and studying linear secret sharing scheme. In this paper, we show that in the binary cyclic code all of the codewords are minimal, except 0 and 1. Then, we obtain a result about the number of minimal codewords in the binary cyclic codes.
基金supported in part by Fundamental Research Fund for the Central Universities of ChinaZhejiang Provincial Natural Science Foundation(Grant No.LQ12A01019)+2 种基金National Natural Science Foundation of China(Grant No.11201418)Research Fund for Doctoral Programs from the Ministry of Education of China(Grant No.20120101120089)supported in part by AcRF grant(Grant No.R-146-000-158-112)
文摘In this paper, we make some progress towards a well-known conjecture on the minimum weights of binary cyclic codes with two primitive nonzeros. We also determine the Walsh spectrum of Tr(x d ) over F2m in the case where m = 2t, d = 3 + 2t+1 and gcd(d, 2m-1) = 1.
