Higher weight spectra and Betti numbers of Reed-Muller codes $RM_q(2,2)$

TL;DR

This paper determines the higher weight spectra and Betti numbers of Reed-Muller codes RM_q(2,2) for all prime powers q, linking code properties to algebraic and combinatorial structures.

Contribution

It explicitly computes the higher weight spectra and Betti numbers of Reed-Muller codes RM_q(2,2) for all prime powers q, using connections to Stanley-Reisner rings and matroid theory.

Findings

Higher weight spectra are explicitly determined for all q.

Betti numbers of associated matroids are explicitly computed.

Connections established between code weights and algebraic invariants.

Abstract

We determine the higher weight spectra of $q$-ary Reed-Muller codes $C_q=RM_q(2,2)$ for all prime powers $q$. This is equivalent to finding the usual weight distributions of all extension codes of $C_q$ over every field extension of $F_q$ of finite degree. To obtain our results we will utilize well-known connections between these weights and properties of the Stanley-Reisner rings of a series of matroids associated to each code $C_q$. In the process, we are able to explicitly determine all the graded Betti numbers of matroids associated to $C_q$ and its elongations.