Isometry Groups of Combinatorial Codes

TL;DR

This paper investigates the automorphism and monomial automorphism groups of combinatorial codes, revealing they can differ significantly, and provides characterizations and constructions for codes with specific isometry groups.

Contribution

It characterizes the two isometry groups for combinatorial codes and constructs codes with prescribed automorphism and monomial automorphism groups.

Findings

Automorphism and monomial automorphism groups can differ arbitrarily in combinatorial codes.

Existence of codes with full automorphism group and trivial monomial automorphism group.

Characterization and construction methods for codes with specific isometry groups.

Abstract

Two isometry groups of combinatorial codes are described: the group of automorphisms and the group of monomial automorphisms, which is the group of those automorphisms that extend to monomial maps. Unlike the case of classical linear codes, where these groups are the same, it is shown that for combinatorial codes the groups can be arbitrary different. Particularly, there exist codes with the full automorphism group and the trivial monomial automorphism group. In the paper the two groups are characterized and codes with predefined isometry groups are constructed.