New Optimal Variable-Weight Optical Orthogonal Codes

Abstract

Let $W, \ L$, and Q denote the sets {w ₀, w ₁, ..., w _p }, $\{\lambda_a^0,$ $\lambda_a^1, \ldots, \lambda_a^p\}$ and {q ₀, q ₁, ..., q _p }, respectively. An (n, W, L, λ _c , Q) variable-weight optical orthogonal code C, or (n, W, L, λ _c , Q)-OOC, is a collection of binary n-tuples such that for each 0 ≤ i ≤ p, there are exactly q _i |C| codewords of weight w _i , L is related to periodic auto-correlation, and λ _c is related to periodic cross-correlation. The notation (n, W, λ, Q)- OOC is used to denote an (n, W, L, λ _c , Q)-OOC with the property that $\lambda_a^0=\lambda_a^1=\ldots=\lambda_a^p=\lambda_c=\lambda$. An (n, W, L, λ _c , Q)-OOCs was introduced by Yang for multimedia optical CDMA systems with multiple quality of service (QoS) requirements. A cyclic (v,K, 1) difference family (cyclic (v,K, λ)-DF in short) is a family $\cal F=\{B_1, B_2, \ldots, B_t\}$ of t subsets of Z _v , the residue ring of integers modulo v, K = {|B _i |: 1 ≤ i ≤ t}, such that the differences in $\cal F$, $\Delta \cal F=\bigcup_{B\in \cal F}\Delta B$ cover each nonzero element of Z _v exactly λ times, where for each $B\in \cal F$, $\Delta B=\{x-y: x, y\in B, x\ne y\}$, and $|dev \ B_i|=v$, 1 ≤ i ≤ t, $dev \ B_i=\{B_i+g: g\in Z_v\}$. A cyclic (v,W, 1,Q)-DF is defined to be a cyclic (v,W, 1)-DF with the property that the fraction of number of blocks of size w _i is q _i , 0 ≤ i ≤ p. In this paper, constructions for cyclic (v, {4, 6, 7},1,{1/3, 1/3, 1/3})-DFs for primes $v\equiv 1\pmod {84}$, (v, {4, u},1,{1/2, 1/2})-DFs for primes $v\equiv 1\pmod {u(u-1)+12}$, $u\equiv 0, 1\pmod 3>4$ are presented. New optimal (v, W, 1,Q)-OOCs for 2 ≤ |W| ≤ 4 are then obtained.