A class of quaternary linear codes improving known minimum distances

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).