Optical Orthogonal Signature Pattern Codes With Maximum Collision Parameter $2$ and Weight $4$

Combinatorial constructions of optimal (m, n, 4, 2) optical orthogonal signature pattern codes

“…Theorem 2.1 [45] An (m, n, w, λ)-OOSPC of size u is equivalent to a strictly Z m × Z n -invariant (λ + 1)-(mn, w, 1) packing design having u base blocks.…”

“…Lemma 2.2 [45] Let m and n be positive integers. If mn ≡ 0 (mod 24) then Θ(m, n, 4, 2) ≤ J(m, n, 4, 2) − 1.…”

“…Sawa [45] also posed an open problem: Does there exist an optimal (6, n, 4, 2)-OOSPC attaining the Johnson bound (1.1) for a positive integer n, not being a multiple of 4 in general? By analyzing the leave of a strictly Z m × Z n -invariant PQS(mn), we show that there does not exist an (m, n, 4, 2)-OOSPC attaining the upper bound (1.1) for m, n ≡ 0 (mod 3) with mn ≡ 18, 36 (mod 72).…”

Determination of the sizes of optimal (m,n,k,λ,k−1)-OOSPCs for Pan1, Chang2 2013Discrete Mathematics

“…They have also given a new upper bound of Θ(m, n, 3, 2, 1) and have presented two algebraic constructions for optimal (m, n, 3, 2, 1)-OOSPC. In [5], Sawa has shown an equivalence relation between optimal OOSPC and strictly invariant packing design and has constructed an optimal (2 ε x, y, 4, 2)-OOSPC with ε ∈ {1, 2} and x, y are positive integers, whose each factor being a prime less than 500 000 and congruent to 53 or 77 modulo 120 or belonging to S = {5, 13, 17, 25, 29, 37, 41, 53, 61, 85, 89, 97, 101, 113, 137, 149, 157, 169, 173, 193, 197, 229, 233, 289, 293, 317}. In the remainder of this paper, we shall not treat the construction of optimal OOSPCs, but instead, we focus our attention on two classes of OOSPCs and obtain the formulas of Θ(m, n, k, k, k − 1) and Θ(m, n, k, k − 1).…”

Optimal optical orthogonal signature pattern codes with weight three and cross-correlation constraint one

“…The exact number of codewords of an optimal (m, n, 3, λ a , 1)-OOSPC is determined for any positive integers m, n ≡ 2 (mod 4) and λ a ∈ {2, 3}.When m and n are coprime, it has been shown in [28] that an (m, n, k, λ a , λ c )-OOSPC is equivalent to a 1-dimensional (mn, k, λ a , λ c )-optical orthogonal code (OOC). See [1-4, 7, 10, 13, 14, 29] and the references therein for more details on OOCs.When m and n are not coprime, various OOSPCs have been constructed via algebraic and combinatorial methods for the case of λ a = λ c (see [5,6,15,[23][24][25][26]28]). We only quote the following result for later use.…”