Submodule codes as spherical codes in buildings

TL;DR

This paper introduces a new class of codes based on modules over finite chain rings, connecting them to spherical codes in buildings and demonstrating their optimality using combinatorial methods.

Contribution

It generalizes subspace codes via modules over chain rings, establishing a link with Bruhat-Tits buildings and spherical codes, and proves their optimality.

Findings

Defined Sperner codes and proved their optimality in certain cases

Established a connection between these codes and Bruhat-Tits buildings

Showed how the codes serve as analogues of spherical codes in Euclidean space

Abstract

We give a generalization of subspace codes by means of codes of modules over finite commutative chain rings. We define a new class of Sperner codes and use results from extremal combinatorics to prove the optimality of such codes in different cases. Moreover, we explain the connection with Bruhat-Tits buildings and show how our codes are the buildings' analogue of spherical codes in the Euclidean sense.