Maximal (v, k, 2, 1) Optical Orthogonal Codes with k = 6 and 7 and Small Lengths

Abstract

Optical orthogonal codes (OOCs) are used in optical code division multiple access systems to allow a large number of users to communicate simultaneously with a low error probability. The number of simultaneous users is at most as big as the number of codewords of such a code. We consider (v,k,2,1)-OOCs, namely OOCs with length v, weight k, auto-correlation 2, and cross-correlation 1. An upper bound B0(v,k,2,1) on the maximal number of codewords of such an OOC was derived in 1995. The number of codes that meet this bound, however, is very small. For k≤5, the (v,k,2,1)-OOCs have already been thoroughly studied by many authors, and new upper bounds were derived for (v,4,2,1) in 2011, and for (v,5,2,1) in 2012. In the present paper, we determine constructively the maximal size of (v,6,2,1)- and (v,7,2,1)-OOCs for v≤165 and v≤153, respectively. Using the types of the possible codewords, we calculate an upper bound B1(v,k,2,1)≤B0(v,k,2,1) on the code size of (v,6,2,1)- and (v,7,2,1)-OOCs for each length v≤720 and v≤340, respectively.