Linear Codes Associated to Symmetric Determinantal Varieties: Even Rank Case

TL;DR

This paper studies linear codes from symmetric determinantal varieties over finite fields, deriving formulas for code weights, computing minimum distances for even rank bounds, and providing tables for small cases.

Contribution

It introduces a formula for code word weights, computes minimum distances for codes with even rank bounds, and offers corrected and extended tables for small matrix sizes.

Findings

Derived a weight formula for code words.

Computed minimum distances for codes with even rank bounds.

Provided tables for symmetric matrices up to size 5.

Abstract

We consider linear codes over a finite field of odd characteristic, derived from determinantal varieties, obtained from symmetric matrices of bounded ranks. A formula for the weight of a code word is derived. Using this formula, we have computed the minimum distance for the codes corresponding to matrices upper-bounded by any fixed, even rank. A conjecture is proposed for the cases where the upper bound is odd. At the end of the article, tables for the weights of these codes, for spaces of symmetric matrices up to order $5$, are given. We also correct typographical errors in Proposition 1.1/3.1 of [3], and in the last table, and we have rewritten Corollary 4.9 of that paper, and the usage of that Corollary in the proof of Proposition 4.10.