Constructions of optimal balanced $ (m, n, \{4, 5\}, 1) $-OOSPCs

Abstract

Kitayama proposed a novel OCDMA (called spatial CDMA) for parallel transmission of 2-D images through multicore fiber. Optical orthogonal signature pattern codes (OOSPCs) play an important role in this CDMA network. Multiple-weight (MW) optical orthogonal signature pattern code (OOSPC) was introduced by Kwong and Yang for 2-D image transmission in multicore-fiber optical code-division multiple-access (OCDMA) networks with multiple quality of services (QoS) requirements. Some results had been done on optimal balanced (m, n, {3, 4}, 1)-OOSPCs. In this paper, it is proved that there exist optimal balanced (2u, 16v, {4, 5}, 1)-OOSPCs for odd integers u ≥ 1, v ≥ 1.

On balanced $(Z_{4u}\times Z_{8v},\{4,5\},1)$ difference packings

“…In this section, we will prove Theorem 1.1. Lemma 4.1 There exists an (u × v, g × h, {4, 5}, 1)-BDP for (u, v, g, h) = (2, 36, 2, 4), (2, 72, 2, 8), (2, 108, 2, 12), (4, 72, 4, 8), (6, 12, 2, 4), (6,36,2,12), (12,24,4,8), (18,4,2,4), (18,12,2,12).…”

“…A (4u × 8, 4 × 8, {4, 5}, 1)-BDP exists from the above, and a (v, 5; 1)-CDM is from Lemma 3.2, the conclusion comes from Corollary 3.4 by using a (4 × 8v, 4 × 8, {4, 5}, 1)-BDP. Lemma 4.8 If u, v are positive integers such that gcd(uv, 30) = 1, then there exists a (gu × hv, g × h, {4, 5}, 1)-BDP for (g, h) = (4, 24), (4, 120), (12,8), (20,24).…”

“…Proof Since a (gu × hv, g × h, {4, 5}, 1)-BDP is equivalent to a (g ′ u × h ′ v, g ′ × h ′ , {4, 5}, 1)-BDP, where (g, h, g ′ , h ′ ) = (4,24,12,8), or (4,120,20,24), then we need only to consider the cases of (g, h) = (4, 24), (4,120).…”

“…-BDP exists from Lemma 4.5, where (g, h) = (4, 8), (4,24), or (12,8). An optimal (4 × 24, {4, 5}, 1)-BDP is equivalent to an optimal (12 × 8, {4, 5}, 1)-BDP, the blocks of an optimal (4 × 24, {4, 5}, 1)-BDP is listed below.…”

“…Optimal (3u×9v, {3, 4}, 1)-BDPs were constructed for any pair of positive integers (u, v) with gcd(u, 3) = 1 [38]. Optimal (2u×16v, {4, 5}, 1)-BDPs were constructed for uv odd [24]. To the authors' knowledge, no other work had been done on optimal (m × n, {4, 5}, 1)-BDPs when m and n are not coprime.…”