Further results on optimal (<italic>m</italic>, <italic>n</italic>, 4, 1) optical orthogonal signature pattern codes

Abstract

(2) mn ≡ 0 (mod 24) 且 gcd(m, n, 6) = 2, 证明 Θ(m, n, 4, 1) = ⌊ mn−1 12 ⌋, 即构造码字容量为 ⌊ mn−1 12 ⌋ 的最优 (m, n, 4, 1)-OOSPC.

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

“…Moreover, if the given balanced (g × h, W, 1)-SP is s × t-regular, then so is the derived balanced (u × v, W, 1)-SP. Construction 2 and its proof are presented in [35], it is an analogue result of Construction 3.8 in [22]. Construction 2.…”

“…When u and v are not coprime, the construction of optimal (u, v, k, λ a , λ c )-OOSPCs becomes difficult. For k = 3, 4, some results have been obtained for optimal (u, v, k, λ a , λ c )-OOSPCs [5], [6], [20], [22], [23], [19], [27], [28]. For general weight k, some results on (asymptotically ) optimal (u, v, k, λ a , λ c )-OOSPCs are also obtained [14], [21], [32].…”

“…{(0,0),(0,1),(0,3),(0,7),(0,12)}, {(0,0),(0,8),(0,18),(0,31),(1,0)}, {(0,0),(0,14),(0,29),(1,1)}, {(0,0),(0,16),(1,2), (1,22)}.…”

On optimal (Z6m×Z6n,4,1) and (Z2m×Z18n,4,1) difference packings and their related codes

“…⋅ Lemma 4.3 (Pan and Chang [26]). Let m n , be positive integers and mn m n 0 (mod 24), gcd( , , 6) = 2 ≡…”

“…The reader can refer to [1,3,[6][7][8]10,31,32] on mn k ( , , 1)-DPs. When m and n are not coprime, to our knowledge the existence of an optimal m n k ( × , , 1)-DP has not been much investigated, the reader can refer to [25][26][27] and the reference therein.…”