Two classes of constacyclic codes with a square-root-like lower bound
Tingfang Chen, Zhonghua Sun, Conghui Xie, Hao Chen, Cunsheng Ding
TL;DR
This paper addresses constructing constacyclic codes with strong minimum distance properties over finite fields and derives two infinite families with square-root-like lower bounds on both $d$ and $d^{\perp}$. It presents a first class based on a split of the negacyclic defining set and a second class built from $q$-weight based defining sets, each yielding explicit dimension formulas and distance bounds; it also constructs two infinite ternary negacyclic self-dual families with comparable bounds. To enrich the code family set, the authors develop several subcodes with additional structure that provide further distance guarantees and self-duality in specific cases, including concrete examples such as a ternary self-dual $[40,20,9]$ code. The results offer distance-optimal or best-known parameter codes with algebraic structure suitable for decoding and implementation, expanding the catalog of practical constacyclic codes. Overall, the work advances the theory and construction of distance-rich, structured codes over finite fields, with potential impact on communication and data integrity applications.
Abstract
Constacyclic codes over finite fields are an important class of linear codes as they contain distance-optimal codes and linear codes with best known parameters. They are interesting in theory and practice, as they have the constacyclic structure. In this paper, an infinite class of $q$-ary negacyclic codes of length $(q^m-1)/2$ and an infinite class of $q$-ary constacyclic codes of length $(q^m-1)/(q-1)$ are constructed and analyzed. As a by-product, two infinite classes of ternary negacyclic self-dual codes with a square-root-like lower bound on their minimum distances are presented.