On symmetric and Hermitian rank distance codes

TL;DR

This paper explores rank distance codes in symmetric and Hermitian matrix spaces over finite fields, constructing larger codes than known bounds and providing new bounds and examples for specific parameters.

Contribution

It constructs new rank distance codes in symmetric and Hermitian matrix spaces that surpass existing additive bounds and establishes new size bounds for certain parameters.

Findings

Existence of large Hermitian codes for even n/odd n/2.

Construction of a 2-code of size q^4+q^3+1 for n=3, q>2.

First infinite family of symmetric 2-codes exceeding known bounds.

Abstract

Let $\cal M$ denote the set ${\cal S}_{n, q}$ of $n \times n$ symmetric matrices with entries in ${\rm GF}(q)$ or the set ${\cal H}_{n, q^2}$ of $n \times n$ Hermitian matrices whose elements are in ${\rm GF}(q^2)$. Then $\cal M$ equipped with the rank distance $d_r$ is a metric space. We investigate $d$-codes in $({\cal M}, d_r)$ and construct $d$-codes whose sizes are larger than the corresponding additive bounds. In the Hermitian case, we show the existence of an $n$-code of $\cal M$, $n$ even and $n/2$ odd, of size $\left(3q^{n}-q^{n/2}\right)/2$, and of a $2$-code of size $q^6+ q(q-1)(q^4+q^2+1)/2$, for $n = 3$. In the symmetric case, if $n$ is odd or if $n$ and $q$ are both even, we provide better upper bound on the size of a $2$-code. In the case when $n = 3$ and $q>2$, a $2$-code of size $q^4+q^3+1$ is exhibited. This provides the first infinite family of $2$-codes of symmetric matrices whose size is larger than the largest possible additive $2$-code.