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- W3105201810 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." @default.
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- W3105201810 date "2020-11-13" @default.
- W3105201810 modified "2023-09-27" @default.
- W3105201810 title "On symmetric and Hermitian rank distance codes." @default.
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