埃尔米特秩距离码

摘要：

令$X=X(n,q)$是$\mathbb{F}_{q^2}$上的$n\times n$厄米特矩阵的集合。众所周知，$X$给出了度量平移关联方案，其类由秩度量导出。我们研究了这个方案中的$d$ -码，即$X$的子集$Y$，它具有这样的性质：对于所有不同的$A,B\in Y$，$A-B$的秩至少是$d$ .我们证明了一个$d$ -码的大小的界限，并表明，在一定的条件下，一个$d$ -码的内部分布是由它的参数。除了$n$和$d$都是偶数且$4\le d\le n-2$外，给出了$d$ -码的构造，它们是$(X,+)$的子群$d$ -码中的最优码.这项工作补充了以前获得的结果，其他几种类型的矩阵在有限域。

英文摘要：

Let $X=X(n,q)$ be the set of $n\times n$ Hermitian matrices over $\mathbb{F}_{q^2}$. It is well known that $X$ gives rise to a metric translation association scheme whose classes are induced by the rank metric. We study $d$-codes in this scheme, namely subsets $Y$ of $X$ with the property that, for all distinct $A,B\in Y$, the rank of $A-B$ is at least $d$. We prove bounds on the size of a $d$-code and show that, under certain conditions, the inner distribution of a $d$-code is determined by its parameters. Except if $n$ and $d$ are both even and $4\le d\le n-2$, constructions of $d$-codes are given, which are optimal among the $d$-codes that are subgroups of $(X,+)$. This work complements results previously obtained for several other types of matrices over finite fields.

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