New Constructions of Binary Cyclic Codes with Both Relatively Large Minimum Distance and Dual Distance
Lingqi Zheng, Weijun Fang, Rongxing Qiu
TL;DR
The paper addresses constructing binary cyclic codes of length $n=2^m-1$ with dimension near $n/2$ that simultaneously achieve large minimum distance $d$ and dual distance $d^{\perp}$. It delivers three construction paradigms based on the parity and factorization of $m$, including two codes for even $m$ with $d, d^{\perp}$ up to $2^{m/2}$, and two families for odd $m$ with $d$ and $d^{\perp}$ achieving $2^{(m+1)/2}$ or close to $2^{(m+3)/2}$ while sustaining large dual distances. For composite $m$ ($m=p_1p_2$), the authors obtain codes with $d$ exceeding $n/\log_2 n$ and $d^{\perp}$ bounded below by a power of two, yielding near-optimal $d\cdot d^{\perp}$ performance; for odd $m$ they present two families whose $d\cdot d^{\perp}$ asymptotically approaches $2n$. The work thereby surpasses several prior bounds (excluding punctured binary Reed-Muller codes) and raises open questions about the maximal possible asymptotic product $d\cdot d^{\perp}$, inviting further exploration of innovative defining-set constructions.
Abstract
Binary cyclic codes are worth studying due to their applications and theoretical importance. It is an important problem to construct an infinite family of cyclic codes with large minimum distance $d$ and dual distance $d^{\perp}$. In recent years, much research has been devoted to improving the lower bound on $d$, some of which have exceeded the square-root bound. The constructions presented recently seem to indicate that when the minimum distance increases, the minimum distance of its dual code decreases. In this paper, we focus on the new constructions of binary cyclic codes with length $n=2^m-1$, dimension near $n/2$, and both relatively large minimum distance and dual distance. For $m$ is even, we construct a family of binary cyclic codes with parameters $[2^m-1,2^{m-1}\pm1,d]$, where $d\ge 2^{m/2}-1$ and $d^\perp\ge2^{m/2}$. Both the minimum distance and the dual distance are significantly better than the previous results. When $m$ is the product of two distinct primes, we construct some cyclic codes with dimensions $k=(n+1)/2$ and $d>\frac{n}{\log_2n},$ where the lower bound on the minimum distance is much larger than the square-root bound. For $m$ is odd, we present two families of binary $[2^m-1,2^{m-1},d]$ cyclic codes with $d\ge2^{(m+1)/2}-1$, $d^\perp\ge2^{(m+1)/2}$ and $d\ge2^{(m+3)/2}-15$, $d^\perp\ge2^{(m-1)/2}$ respectively, which leads that $d\cdot d^\perp$ can reach $2n$ asymptotically. To the best of our knowledge, except for the punctured binary Reed-Muller codes, there is no other construction of binary cyclic codes that reaches this bound.