Several families of binary cyclic codes with good parameters

Abstract

Binary odd-like duadic codes have parameters [n,(n+1)/2], where n is odd. It is well known that the minimum odd weight of odd-like duadic codes has the lower bound n. The binary quadratic-residue codes and the punctured binary Reed-Muller codes of certain order are two families of binary cyclic codes with parameters [n,(n+1)/2,d≥n]. It is very hard to construct an infinite family of binary cyclic codes with length n and dimension near (n+1)/2 whose minimum distances have a square-root bound. Recently, Tang and Ding constructed an infinite family of binary cyclic codes with parameters [2m−1,2m−1,d] whose minimum distances have lower bounds close to the square-root bound. The objective of this paper is to construct several infinite families of binary cyclic codes of length 2m−1 and dimension near 2m−1 whose minimum distance d much exceeds the square-root bound. For m≥9 being odd, an infinite family of binary cyclic codes with parameters [2m,2m−1,d≥3⋅2m−12−1] is presented. For m≥8 being even, an infinite family of binary cyclic codes with parameters [2m,2m−1−1+m2,d≥3⋅2m−22+1] is presented.