2020/11/13 by Antonio Cossidente, Giuseppe Marino, Cossidente, Antonio +3 · 1 citation
Computer Science · Mathematics · #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #cs.IT #math.CO #math.IT
paper · pdf · doi:10.48550/arxiv.2011.06942
arxiv created 2020/11/13 · arxiv updated 2020/11/16
Let \cal M denote the set \cal Sn, q of n × n symmetric matrices with entries in \rm GF(q) or the set \cal Hn, q2 of n × n Hermitian matrices whose elements are in \rm GF(q2). Then \cal M equipped with the rank distance dr is a metric space. We investigate d-codes in (\cal 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 \cal M, n even and n/2 odd, of size (3qn-qn/2)/2, and of a 2-code of size q6+ q(q-1)(q4+q2+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 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.