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

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 B 0 ( 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 B 1 ( v , k , 2 , 1 ) ≤ B 0 ( 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.