Biometrical Letters vol. 46(2), 2009, pp. 89-102
|
OPTIMAL OPTICAL ORTHOGONAL SIGNATURE PATTERN CODES OF WEIGHT 3 Masanori Sawa1, Sanpei Kageyama2 1Nagoya University, Graduate School of Information Science, Nagoya 464-8601, Japan, e-mail: sawa@is.nagoya-u.ac.jp 2Hiroshima Institute of Technology, Department of Environmental Design, Hiroshima 731-5193, Japan, e-mail: s.kageyama.4b@it-hiroshima.ac.jp |
The purpose of this paper is to investigate the existence of optical orthogonal signature pattern codes (OOSPCs) of weight 3 and cross-correlation constraint 1 and to discuss what is optimal for such OOSPCs. First we focus on OOSPCs of auto-correlation constraint 1, where the optimality is decided by the classical upper bound for the number of codewords in an OOSPC, as presented by Kitayama (1994). We provide two constructions - one is a direct construction using certain combinatorial objects, called Skolem sequences, and the other is a recursive construction. Using these constructions, for any odd integers m, n such that either m or n is not congruent to 5 modulo 6, we obtain an optimal OOSPC of size m×n which attains the Kitayama bound. Next we focus on OOSPCs of auto-correlation constraint 2. We prove that the number of codewords in such an OOSPC of size m × n is bounded above by mn/4 or b(mn - 1)/4c OOSPs, according as mn is divisible by 4 or not. This new bound represents a significant improvement on the Kitayama bound. Finally we construct many optimal OOSPCs which attain the new bound, by presenting two new algebraic constructions.
optical orthogonal signature pattern code (OOSPC), packing design, Kitayama bound (Kwong-Yang bound).
