An Infinite Family of Binary Cyclic Codes With Best Parameters

Binary cyclic codes with parameters $[n,(n+1)/2, d\geq \sqrt {n}]$ 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 ${\mathcal {C}}(m)$ with parameters $[2^{m}-1,2^{m-1},d]$ whose minimum distance $d$ much exceeds the square-root bound when $m \geq 11$ is a prime. The binary BCH code ${\mathcal {C}}(3)$ is the binary Hamming code and distance-optimal. The binary BCH code ${\mathcal {C}}(5)$ has parameters $[{31,16,7}]$ and is distance-almost-optimal. The binary BCH code ${\mathcal {C}}(7)$ 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 ${\mathcal {C}}(m)$ with parameters $[2^{m}-1,2^{m-1}]$ for any odd prime $m$ .