Invariant codes, difference schemes, and distributive quasigroups

Abstract

Regard $T=\{(g,\ldots ,g)\in {\mathbb {F}}_{q}^{n}: \, g \in {\mathbb {F}}_q\}$ as an additive subgroup of the vector space ${\mathbb {F}}_q^n$ over a finite field ${\mathbb {F}}_q$. A code ${\mathcal {C}} \subset {\mathbb {F}}_q^n$ is T-invariant if T acts on ${\mathcal {C}}$ induced by the addition of ${\mathbb {F}}_q^n$. Using linear algebra and finite fields results, we investigate which important families of codes are T-invariant, including perfect codes, first-order Reed–Muller codes, and some Reed–Solomon codes. In particular, we characterize which perfect linear codes are T-invariant. A version of the Gilbert–Varshamov theorem for T-invariant codes is presented. We characterize which Hadamard codes are T-invariant for nonlinear codes. As an application, constructions of difference schemes are obtained using our results. Finally, difference schemes of strength 2 over a finite field are characterized using quasigroups endowed with a distributive law.