An Infinite Family of Binary Cyclic Codes With Best Parameters

Binary cyclic codes with parameters [n,(n+1)/2, d ≥ √ n] are very interesting, as their minimum distances have a square-root bound. The binary quadratic residue codes and the punctured binary Reed-Muller codes of order (m-1)/2 for odd m are two infinite families of binary cyclic codes with such parameters. The objective of this paper is to present and analyse an infinite family of binary BCH codes C(m) with parameters [2^m-1,2^m-1,d] whose minimum distance d much exceeds the square-root bound when m ≥ 11 is a prime. The binary BCH code C(3) is the binary Hamming code and distance-optimal. The binary BCH code C(5) has parameters [31,16,7] and is distance-almost-optimal. The binary BCH code C(7) has parameters [127,64,21] and has the best known parameters. In addition, there is no known [2^m-1,2^m-1] binary cyclic code whose minimum distance is better than the minimum distance of this binary BCH code C(m) with parameters [2^m-1,2^m-1] for any odd prime m.