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Register a new userCombinatorial Constructions of Optimal $(m, n,4,2)$ Optical Orthogonal Signature Pattern Codes
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Lijun Ji
Publication date
2015
fields
Informatics Engineering
and research's language is
English
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Optical orthogonal signature pattern codes (OOSPCs) play an important role in a novel type of optical code-division multiple-access (CDMA) network for 2-dimensional image transmission. There is a one-to-one correspondence between an $(m, n, w, lambda)$-OOSPC and a $(lambda+1)$-$(mn,w,1)$ packing design admitting an automorphism group isomorphic to $mathbb{Z}_mtimes mathbb{Z}_n$. In 2010, Sawa gave the first infinite class of $(m, n, 4, 2)$-OOSPCs by using $S$-cyclic Steiner quadruple systems. In this paper, we use various combinatorial designs such as strictly $mathbb{Z}_mtimes mathbb{Z}_n$-invariant $s$-fan designs, strictly $mathbb{Z}_mtimes mathbb{Z}_n$-invariant $G$-designs and rotational Steiner quadruple systems to present some constructions for $(m, n, 4, 2)$-OOSPCs. As a consequence, our new constructions yield more infinite families of optimal $(m, n, 4, 2)$-OOSPCs. Especially, we shall see that in some cases an optimal $(m, n, 4, 2)$-OOSPC can not achieve the Johnson bound.
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Optical orthogonal signature pattern codes (OOSPCs) have attracted wide attention as signature patterns of spatial optical code division multiple access networks. In this paper, an improved upper bound on the size of an $(m,n,3,lambda_a,1)$-OOSPC with $lambda_a=2,3$ is established. The exact number of codewords of an optimal $(m,n,3,lambda_a,1)$-OOSPC is determined for any positive integers $m,nequiv2 ({rm mod } 4)$ and $lambda_ain{2,3}$.
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Optimal rank-metric codes in Ferrers diagrams can be used to construct good subspace codes. Such codes consist of matrices having zeros at certain fixed positions. This paper generalizes the known constructions for Ferrers diagram rank-metric (FDRM) codes. Via a criteria for linear maximum rank distance (MRD) codes, an explicit construction for a class of systematic MRD codes is presented, which is used to produce new optimal FDRM codes. By exploring subcodes of Gabidulin codes, if each of the rightmost $delta-1$ columns in Ferrers diagram $cal F$ has at least $n-r$ dots, where $r$ is taken in a range, then the conditions that an FDRM code in $cal F$ is optimal are established. The known combining constructions for FDRM code are generalized by introducing the concept of proper combinations of Ferrers diagrams.
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As an important coding scheme in modern distributed storage systems, locally repairable codes (LRCs) have attracted a lot of attentions from perspectives of both practical applications and theoretical research. As a major topic in the research of LRCs, bounds and constructions of the corresponding optimal codes are of particular concerns. In this work, codes with $(r,delta)$-locality which have optimal minimal distance w.r.t. the bound given by Prakash et al. cite{Prakash2012Optimal} are considered. Through parity check matrix approach, constructions of both optimal $(r,delta)$-LRCs with all symbol locality ($(r,delta)_a$-LRCs) and optimal $(r,delta)$-LRCs with information locality ($(r,delta)_i$-LRCs) are provided. As a generalization of a work of Xing and Yuan cite{XY19}, these constructions are built on a connection between sparse hypergraphs and optimal $(r,delta)$-LRCs. With the help of constructions of large sparse hypergraphs, the length of codes constructed can be super-linear in the alphabet size. This improves upon previous constructions when the minimal distance of the code is at least $3delta+1$. As two applications, optimal H-LRCs with super-linear length and GSD codes with unbounded length are also constructed.
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