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On symmetric and Hermitian rank distance codes
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 M denote the set Sn,q of n×n symmetric matrices with entries in Fq or the set Hn,q2 of n×n Hermitian matrices whose elements are in Fq2. Then M equipped with the rank distance dr is a metric space. We investigate d–codes in (M,dr) 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 M, n even and n/2 odd, of size (3qnqn/2)/2, and of a 2–code of size q6+q(q1)(q4+q2+1)/2, for n=3. In the symmetric case, if n is odd, 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 q4+q3+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 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

表示集合 小号n,qn×n 具有条目的对称矩阵 Fq 或集合 Hn,q2n×n Hermitian 矩阵,其元素在 Fq2. 然后配备秩距离 dr是度量空间。我们研究d -codes(,dr)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n/2 奇怪的,大小 (3qn-qn/2)/2, 和一个 2 码的大小 q6+q(q-1)(q4+q2+1)/2, 为了 n=3. 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n=3q>2, 大小为 2 的代码 q4+q3+1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]。

更新日期:2022-01-24
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