A Class of Binary Cyclic Codes and Their Weight Distributions

Chao HE  Rong LUO  Mei YANG  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E103-A   No.3   pp.634-637
Publication Date: 2020/03/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.2019EAL2132
Type of Manuscript: LETTER
Category: Coding Theory
Keyword: 
linear code,  cyclic code,  binary code,  Gauss period,  weight distribution,  

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Summary: 
Let m, k be positive integers with m=2k and k≥3. Let C(u, ν) is a class of cyclic codes of length 2m-1 whose parity-check polynomial is mu(x)mν(x), where mu(x) and mν(x) are the minimal polynomials of α-u and α over GF(2). For the case $(u, u)=(1, rac{1}{3}(2^m-1))$, the weight distributions of binary cyclic codes C(u, ν) was determined in 2017. This paper determines the weight distributions of the binary cyclic codes C(u, ν) for the case of (u, ν)=(3, 2k-1+1). The application of these cyclic codes in secret sharing is also considered.