New optimal variable-weight optical orthogonal codes

  • Authors:

  • Dianhua Wu;Jiayun Cao;Pingzhi Fan

  • Affiliations:

  • Keylab of Information Coding and Transmission, Lab of Traffic Information Engineering and Control, Southwest Jiaotong University, Chengdu, China;Keylab of Information Coding and Transmission, Lab of Traffic Information Engineering and Control, Southwest Jiaotong University, Chengdu, China;Keylab of Information Coding and Transmission, Lab of Traffic Information Engineering and Control, Southwest Jiaotong University, Chengdu, China

  • Venue:
  • SETA'10 Proceedings of the 6th international conference on Sequences and their applications
  • Year:
  • 2010

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Abstract

Let W, L, and Q denote the sets {w0, w1, ..., wp}, {λa0, λa1, ..., λap} and {q0, q1, ..., qp}, respectively. An (n, W, L, λc, Q) variable-weight optical orthogonal code C, or (n, W, L, λc, Q)-OOC, is a collection of binary n-tuples such that for each 0 ≤ i ≤ p, there are exactly qi|C| codewords of weight wi, L is related to periodic auto-correlation, and λc is related to periodic cross-correlation. The notation (n, W, L, λ, Q)- OOC is used to denote an (n, W, L, λc, Q)-OOC with the property that λa0 = λa1 = ... = λap = λc = λ. An (n, W, L, λc, 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 F = {B1, B2, ..., Bt} of t subsets of Zv, the residue ring of integers modulo v, K = {|Bi| : 1 ≤ i ≤ t}, such that the differences in F, ΔF = ∪B∈F ΔB cover each nonzero element of Zv exactly λ times, where for each B ∈ F, ΔB = {x - y : x, y ∈ B, x ≠ y}, and |dev Bi| = v, 1 ≤ i ≤ t, dev Bi = {Bi + g : g ∈ Zv}. 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 wi is qi, 0 ≤ i ≤ p. In this paper, constructions for cyclic (v, {4, 6, 7}, 1, {1/3, 1/3, 1/3})-DFs for primes v ≡ 1 (mod 84), (v, {4, u}, 1, {1/2, 1/2})-DFs for primes v ≡ 1 (mod u(u-1) + 12), u ≡ 0, 1 (mod 3) 4 are presented. New optimal (v, W, 1, Q)-OOCs for 2 ≤ |W| ≤ 4 are then obtained.