Geometric representations of linear codes

TL;DR

This paper introduces a geometric way to represent certain linear codes using simplicial complexes, characterizes which codes are representable over specific fields, and explores implications for code properties and applications.

Contribution

It establishes that linear codes over rationals and GF(p) are triangular representable, links this to weight enumerators, and identifies limitations for codes over other fields.

Findings

Linear codes over rationals and GF(p) are triangular representable.

Triangular representation determines the weight enumerator of the code.

Some codes over other fields are not triangular representable.

Abstract

We say that a linear code C over a field F is triangular representable if there exists a two dimensional simplicial complex $\Delta$ such that C is a punctured code of the kernel ker $\Delta$ of the incidence matrix of $\Delta$ over F and there is a linear mapping between C and ker $\Delta$ which is a bijection and maps minimal codewords to minimal codewords. We show that the linear codes over rationals and over GF(p), where p is a prime, are triangular representable. In the case of finite fields, we show that this representation determines the weight enumerator of C. We present one application of this result to the partition function of the Potts model. On the other hand, we show that there exist linear codes over any field different from rationals and GF(p), p prime, that are not triangular representable. We show that every construction of triangular representation fails on a very weak condition that a linear code and its triangular representation have to have the same dimension.