Determination of Sizes of Optimal Three‐Dimensional Optical Orthogonal Codes of Weight Three with the AM‐OPP Restriction

Abstract

In this paper, we further investigate the constructions on three-dimensional (u × v × w, k, 1) optical orthogonal codes with the at most one optical pulse per wavelength/time plane restriction (briefly AM-OPP 3-D (u × v × w, k, 1)-OOCs) by way of the corresponding designs. Several new auxiliary designs such as incomplete holey group divisible designs and incomplete group divisible packings are introduced and therefore new constructions are presented. As a consequence, the exact number of codewords of an optimal AM-OPP 3-D (u × v × w, 3, 1)-OOC is finally determined for any positive integers v, w and u ≥ 3.

Some progress on optimal $ 2 $-D $ (n\times m,3,2,1) $-optical orthogonal codes

“…Lemma 2.5. ( [35]) An m-cyclic 3-GDD of type (vm) u exists if and only if (1) when u = 3, m is odd, or m is even and v is even; (2) when u ≥ 4, (u − 1)vm ≡ 0 (mod 2), u(u − 1)vm ≡ 0 (mod 3), and v ≡ 0 (mod 2) if u ≡ 2, 3 (mod 4) and m ≡ 2 (mod 4).…”

“…An m-cyclic k-GDP of type (vm) u is called optimal if it contains the largest possible number of base blocks. The existence of an optimal 3-SCGDP of type m u was solved in [35]. We quote the result when m is odd for later use.…”

Maximum w‐cyclic holey group divisible packings with block size three and applications to optical orthogonal codes

“…Proof By Theorem 1.1, there exists a 3‐HGDD of type with six blocks. Apply Construction 2.6 with a 3‐SCGDP of type with base blocks which exists from [21,22, Theorem 5.18] to obtain a ‐cyclic 3‐HGDP of type with base blocks. So the conclusion follows by Lemma 3.2.…”

The spectrum of semicyclic holey group divisible designs with block size three

“…Construction 2.12 (Wang and Chang [31], Construction 3.19). If there exist a k-SCIHGDD of type n r m ( , , ) t and an l-SCGDD of type w k , then there exists an l-SCIHGDD of type n r mw ( , , ( ) ) t .…”

“…Subsequently, a systematic investigation on SCHGDDs was made in [18,19]. Recently, SCHGDDs have been successfully applied to construct optimal 2D OOCs with better cross-correlation than autocorrelation [16,17,34], optimal 3D OOCs with certain restrictions [13,30,31] and semicyclic 2D balanced sampling plans [19]. Lemma 1.3 (Feng et al [19], Lemma 3.4 and Theorem 4.12).…”