A Class of Binary Cyclic Codes and Their Weight Distributions

Abstract

Let m, k be positive integers with m=2k and k≥3. Let C_{(u, ν)} is a class of cyclic codes of length 2^m-1 whose parity-check polynomial is m_u(x)m_ν(x), where m_u(x) and m_ν(x) are the minimal polynomials of α^-u and α^-ν over GF(2). For the case $(u,\nu)=(1,\frac{1}{3}(2^m-1))$, 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, 2^k-1+1). The application of these cyclic codes in secret sharing is also considered.