Binary [<i>n</i>, (<i>n</i> + 1)/2] Cyclic Codes With Good Minimum Distances
Chunming Tang, Cunsheng Ding
Abstract
The binary quadratic-residue codes and the punctured Reed-Muller codes <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathcal {R}}_{2}((m-1)/2, m))$ </tex-math></inline-formula> are two families of 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> . These two families of binary cyclic codes are interesting partly due to the fact that their minimum distances have a square-root bound. The objective of this paper is to construct two families of binary cyclic codes of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{m}-1$ </tex-math></inline-formula> and dimension near <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{m-1}$ </tex-math></inline-formula> with good minimum distances. When <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m \geq 3$ </tex-math></inline-formula> is odd, the codes become a family of duadic 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">$[2^{m}-1, 2^{m-1}, d]$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d \geq 2^{(m-1)/2}+1$ </tex-math></inline-formula> if <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m \equiv 3 \pmod {4}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d \geq 2^{(m-1)/2}+3$ </tex-math></inline-formula> if <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m \equiv 1 \pmod {4}$ </tex-math></inline-formula> . The two families of binary cyclic codes contain some optimal binary cyclic codes.