Completely regular codes in graphs covered by a Hamming graph

Abstract

In Cayley graphs on the additive group of a small vector space over GF(q), q=2,3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=2,3$$\end{document}, we look for completely regular (CR) codes whose parameters are new in Hamming graphs over the same field. The existence of a CR code in such a Cayley graph G implies the existence of a CR code with the same parameters in the corresponding Hamming graph that covers G. In such a way, we find several completely regular codes with new parameters in Hamming graphs over GF(2) and GF(3). The most interesting findings are two new CR-1 (with covering radius 1) codes that are independent sets (such CR are equivalent to optimal orthogonal arrays attaining the Bierbrauer–Friedman bound) and one new CR-2 code. By recursive constructions, every new CR code induces an infinite sequence of CR codes (in particular, optimal orthogonal arrays if the original code is CR-1 and independent). Some of the found ternary CR codes disprove a conjectured generalization of the correlation-immunity bound on the strength of binary orthogonal arrays to the non-binary case. In between, we classify feasible parameters of CR codes in several strongly regular graphs.