Linear codes on posets with extension property

TL;DR

This paper explores linear and additive codes in poset-based spaces with the extension property, analyzing their duality, invariants, and automorphism extensions, with implications for code classification and poset isomorphism problems.

Contribution

It introduces a framework connecting codes, association schemes, and poset duality, and studies invariants and automorphism extension properties in poset codes.

Findings

Dual association schemes are isomorphic iff the poset is self-dual.

Invariants analogous to weight enumerators are linked to classical tree isomorphism.

Automorphisms of ideals extend to automorphisms of the poset in spaces with the extension property.

Abstract

We investigate linear and additive codes in partially ordered Hamming-like spaces that satisfy the extension property, meaning that automorphisms of ideals extend to automorphisms of the poset. The codes are naturally described in terms of translation association schemes that originate from the groups of linear isometries of the space. We address questions of duality and invariants of codes, establishing a connection between the dual association scheme and the scheme defined on the dual poset (they are isomorphic if and only if the poset is self-dual). We further discuss invariants that play the role of weight enumerators of codes in the poset case. In the case of regular rooted trees such invariants are linked to the classical problem of tree isomorphism. We also study the question of whether these invariants are preserved under standard operations on posets such as the ordinal sum and the like.