Cyclic and Negacyclic Codes with Optimal and Best Known Minimum Distances

TL;DR

The paper tackles the problem of constructing distance-optimal and best-known cyclic and negacyclic codes over small finite fields. It develops three core directions: (i) infinite families of distance-optimal binary cyclic codes with $d=6$ and distance-optimal quaternary cyclic codes with $d=4$, (ii) multiple infinite families over $\mathbf{F}_2,\mathbf{F}_3,\mathbf{F}_4,\mathbf{F}_5,\mathbf{F}_7,\mathbf{F}_9$ achieving $d_{ ext{max}}\le d+8$, yielding 145 optimal or best-known codes not equivalent to Grassl's, and (iii) rate-1/2 negacyclic code families with $d\ge c n/\log_q n$, demonstrating substantial asymptotic growth potential. The results leverage BCH-type bounds and cyclotomic coset defining sets to produce explicit, structured algebraic codes with substantial practical encoding/decoding benefits and expanded catalogs of optimal/best-known codes for small fields. Overall, the work significantly enlarges the family of algebraically constructed codes with provable distance properties and complements existing numerical best-known codes.

Abstract

In this paper, we construct a new family of distance-optimal binary cyclic codes with the minimum distance $6$ and a new family of distance-optimal quaternary cyclic codes with the minimum distance $4$. We also construct several families of cyclic and negacyclic codes over ${\bf F}_2$, ${\bf F}_3$, ${\bf F}_4$, ${\bf F}_5$, ${\bf F}_7$ and ${\bf F}_9$ with good parameters $n,\,k,\,d$, such that the maximal possible minimum distance $d_{max}$ of a linear $[n, k]_q$ code is at most $d_{max} \leq d+8$. The first codes in these families have optimal or best known minimum distances. $145$ optimal or best known codes are constructed as cyclic codes, negacyclic codes, their shortening codes and punctured codes. All optimal or best known codes constructed in this paper are not equivalent to the presently best known codes. Several infinite families of negacyclic $[n,\frac{n+1}{2}, d]_q$ codes or $[n, \frac{n}{2}, d]_q$ codes, such that their minimum distances satisfy $d\approx O(\frac{n}{\log_q n})$, are also constructed. These are first several families of such negacyclic codes.