Journal of Combinatorial Theory Series A ( IF 1.1 ) Pub Date : 2022-01-21 , DOI: 10.1016/j.jcta.2022.105597 Antonio Cossidente , Giuseppe Marino , Francesco Pavese
Let denote the set of symmetric matrices with entries in or the set of Hermitian matrices whose elements are in . Then equipped with the rank distance is a metric space. We investigate d–codes in 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 , n even and odd, of size , and of a 2–code of size , for . In the symmetric case, if n is odd, we provide better upper bound on the size of a 2–code. In the case when and , a 2–code of size 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 and an answer to a question posed in [25, Section 7], see also [23, p. 176].
中文翻译:
关于对称和 Hermitian 秩距离码
让 表示集合 的 具有条目的对称矩阵 或集合 的 Hermitian 矩阵,其元素在 . 然后配备秩距离 是度量空间。我们研究d -codes并构造d码,其大小大于相应的加法界限。在 Hermitian 的情况下,我们证明了一个n码的存在, n偶数和 奇怪的,大小 , 和一个 2 码的大小 , 为了 . 在对称情况下,如果n是奇数,我们会为 2 码的大小提供更好的上限。在这种情况下 和 , 大小为 2 的代码 被展出。这提供了第一个无限系列的对称矩阵的 2 码,其大小大于最大可能的加法 2 码,并回答了 [25, Section 7] 中提出的问题,另见 [23, p. 176]。