Constacyclic codes with best-known parameters
Zekai Chen, Min Sha
TL;DR
The work addresses constructing $q$-ary constacyclic codes of length $n$ with dimension near $n/2$ and minimum distance on the order of $c n/\log_q n$ by classifying cyclotomic cosets by size and forming defining sets from the first half of cosets. The authors develop a general framework that achieves a lower bound $d \ge floor(qN/(2(q-1)))$ (and variants) when the defining set uses ceil/floor selections, and they analyze several length forms: prime, $n=(q^p-1)/rs$, $n=\Phi_{p^b}(q)$, and $n=\Phi_{p_1p_2}(q)$, producing infinite families with optimal or best-known parameters. The main contributions include two infinite-coset-size constructions, explicit expressions for coset counts $N_l$, and comprehensive code families that reach or approach Grassl’s best-known distances for diverse lengths. The results offer practical, constructive methods to generate constacyclic codes with strong distance properties, expanding the catalog of codes with provable near-optimal performance for various lengths and field sizes.
Abstract
In this paper, we construct several infinite families of $q$-ary constacyclic codes over a finite field $\mathbb{F}_q$ with length $n$, dimension around $n/2$, and minimum distance at least $cn/\log_q n$ for some positive constant $c$. They contain many constacyclic codes with optimal, or almost-optimal, or best-known parameters. We also consider various forms of the length $n$.