An Infinite Family of Binary Cyclic Codes With Best Parameters
Zhonghua Sun, Chengju Li, Cunsheng Ding
Abstract
Binary cyclic codes with parameters <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$[n,(n+1)/2, d\geq \sqrt {n}]$ </tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(m-1)/2$ </tex-math></inline-formula> for odd <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathcal {C}}(m)$ </tex-math></inline-formula> with parameters <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$[2^{m}-1,2^{m-1},d]$ </tex-math></inline-formula> whose minimum distance <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> much exceeds the square-root bound when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m \geq 11$ </tex-math></inline-formula> is a prime. The binary BCH code <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathcal {C}}(3)$ </tex-math></inline-formula> is the binary Hamming code and distance-optimal. The binary BCH code <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathcal {C}}(5)$ </tex-math></inline-formula> has parameters <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$[{31,16,7}]$ </tex-math></inline-formula> and is distance-almost-optimal. The binary BCH code <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathcal {C}}(7)$ </tex-math></inline-formula> has parameters <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$[{127,64,21}]$ </tex-math></inline-formula> and has the best known parameters. In addition, there is no known <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$[2^{m}-1,2^{m-1}]$ </tex-math></inline-formula> binary cyclic code whose minimum distance is better than the minimum distance of this binary BCH code <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathcal {C}}(m)$ </tex-math></inline-formula> with parameters <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$[2^{m}-1,2^{m-1}]$ </tex-math></inline-formula> for any odd prime <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> .