Combinatorial Constructions Of Optimal (n,3,2,1) Optical Orthogonal Codes

Let n, k, λa, λcbe positive integers. An (n, k, λa, λc) optical orthogonal code C(briefly (n, k, λa, λc)-OOC) is a family of (0,1)-sequences (called codewords) of lengthn, Hamming weight k, auto-correlation constraint λa, cross correlation constraint λc,and the codewords satisfy the auto-correlation and cross correlation.Optical orthogonal code has good correlation properties and has an importantapplication in a fiber-optic code division multiple access (CDMA) channel. In recen-t years, optical orthogonal codes are used to realize multimedia transmission in thelocal area network (LAN) based on fiber-optic channel and high speed CDMA com-munication systems. The optical orthogonal codes research mainly concentrated onthe case λa=λc. When λa> λc, the code capacity is greater than the one in caseλa=λc. In this paper, we study the combinatorial construction of optimal opticalorthogonal codes with length n, Hamming weight k=3, λa=2and λc=1(briefly(n,3,2,1)-OOCs). When n is even, the upper bound of an (n,3,2,1)-OOC are ob-tained by linear programming, and use Skolem type sequences to directly constructoptimal (n,3,2,1)-OOCs. When n is odd, The relation between (n,3,2,1)-OOCand (n,3,2,1) equi-diference cyclic packing (briefly EDCP(n,3,2,1)) is established,then use EDCP(n,3,2,1) to provide some recursive constructions, and apply them toobtain some families of optimal (n,3,2,1)-OOCs.