A class of quaternary linear codes improving known minimum distances

Abstract

For a triple $(F,K,E)$ of finite fields with characteristic $p$ and with $K$ an intermediate field of $E/F$, we describe the $F$-vector space of all mappings from $E$ to $F$ which are constant on $K$-conjugate classes by means of a certain basis. Based on this description, constructions of codes together with an analysis of their designed minimum distances are provided when (a) $E=K$, or when (b) $[E:K]=p$ and $[K:F]$ is a power of $p$. An application to the quaternary field $F=\mathbb {F}_4$ yields a lot of good codes in (a), and in (b) even more than 50 codes which improve the currently best known minimum distances as available from codetables.de (Grassl in Code Tables: Bounds on the Parameters of Various Types of Codes, www.iks.kit.edu/home/grassl/codetables/, 2014).