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Two-Dimensional Optical Orthogonal Codes and Semicyclic Group Divisible Designs

机译：二维光学正交码和半环群可分式设计

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摘要

A $(ntimes m, k, rho)$ two-dimensional optical orthogonal code (2-D OOC), ${cal C}$, is a family of $ntimes m (0, 1)$-arrays of constant weight $k$ such that $sum_{i=1}^{n}sum_{j=0}^{m-1}A(i, j)B(i, joplus_{m}tau)leq rho$ for any arrays $A, B$ in ${cal C}$ and any integer $tau$ except when $A = B$ and $tauequiv 0$ (mod $m$ ), where $oplus_{m}$ denotes addition modulo $m$. Such codes are of current practical interest as they enable optical communication at lower chip rate. To simplify practical implementation, the AM-OPPW (at most one-pulse per wavelen-ngth) restriction is often appended to a 2-D OOC. An AM-OPPW 2-D OOC is optimal if its size is the largest possible. In this paper, the notion of a perfect AM-OPPW 2-D OOC is proposed, which is an optimal $(ntimes m, k, rho)$ AM-OPPW 2-D OOC with cardinality ${m^{rho}n(n-1)cdots (n-rho)over k(k-1)cdots (k-rho)}$ . A link between optimal $(ntimes m, k, rho)$ AM-OPPW 2-D OOCs and block designs is developed. Some new constructions for such optimal codes are described by means of semicyclic group divisible designs. Several new infinite families of perfect $(ntimes m, k, 1)$ AM-OPPW 2-D OOCs with $kin {2,3,4}$ are thus produced.
机译：$ {ntimes m，k，rho）$二维光学正交码（2-D OOC）$ {cal C} $是恒重$ ntimes m（0，1）$数组的一个家庭k $使得任意数组$ sum_ {i = 1} ^ {n} sum_ {j = 0} ^ {m-1} A（i，j）B（i，joplus_ {m} tau）leq rho $ $ {cal C} $中的A，B $和任何整数$ tau $，除非$ A = B $和$ tauequiv 0 $（mod $ m $），其中$ oplus_ {m} $表示加模$ m $。由于这样的代码能够以较低的码片速率进行光通信，因此具有当前的实际意义。为了简化实际实施，通常将AM-OPPW（每个波峰最多一个脉冲）限制附加到2-D OOC。如果AM-OPPW二维OOC的尺寸尽可能大，则是最佳选择。本文提出了一个理想的AM-OPPW二维OOC的概念，它是基数为$ {m ^ {rho} n的最优$（ntimes m，k，rho）$ AM-OPPW二维OOC （n-1）个点（n-rho）在k（k-1）个点（k-rho）} $上。建立了最优的（ntimes m，k，rho）$ AM-OPPW 2-D OOC与模块设计之间的联系。通过半环群可整分设计描述了这种最优代码的一些新结构。这样就产生了几个新的无穷系列，它们具有$ kin {2,3,4} $的完美$（ntimes m，k，1）$ AM-OPPW 2-D OOC。

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