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# New Optimal Variable-Weight Optical Orthogonal Codes

• Dianhua Wu
• Jiayun Cao
• Pingzhi Fan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6338)

## 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.

## Keywords

cyclic difference family difference family optical orthogonal code variable-weight OOC

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## Copyright information

© Springer-Verlag Berlin Heidelberg 2010

## Authors and Affiliations

• Dianhua Wu
• 1
• Jiayun Cao
• 1
• Pingzhi Fan
• 1
1. 1.Keylab of Information Coding and Transmission, Lab of Traffic Information Engineering and ControlSouthwest Jiaotong UniversityChengduChina

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