Linear codes associated to symmetric determinantal varieties; General case

TL;DR

This paper investigates linear codes derived from symmetric determinantal varieties over finite fields, providing explicit formulas for their weight distribution and determining their minimum distance for matrices of bounded rank.

Contribution

It establishes a connection between codeword weights and association scheme Q-numbers, and determines the minimum distance for codes from symmetric matrices of bounded rank.

Findings

Derived explicit weight distribution formulas for these codes.

Determined minimum distances for codes with both odd and even rank bounds.

Connected code properties to algebraic combinatorics via association schemes.

Abstract

The study of linear codes over a finite field of odd cardinality, derived from determinantal varieties obtained from symmetric matrices of bounded rank, was initiated in a recent paper by the authors. There, one found the minimum distance of the code obtained from evaluating homogeneous linear functions at all symmetric matrices with rank, which is, at most, a given even number. Furthermore, a conjecture for the minimum distance of codes from symmetric matrices with ranks bounded by an odd number was given. In this article, we continue the study of codes from symmetric matrices of bounded rank. A connection between the weights of the codewords of this code and Q-numbers of the association scheme of symmetric matrices is established. Consequently, we get a concrete formula for the weight distribution of these codes. Finally, we determine the minimum distance of the code obtained from evaluating homogeneous linear functions at all symmetric matrices with rank at most a given number, both when this number is odd and when it is even.