Applications and Constructions of Optical Codes With Large Cardinality and Multiple Tree Structures

Abstract

Algebraic optical codes with large cardinality and unique tree structures of multiple levels of subsets of codewords for adjustable code performance and cardinality have recently been proposed. As shown in this paper, these characteristics support new network architecture and applications for rapid code switching and physical-layer security in optical code-division multiple access networks. In addition, a new "translate" method of converting "additive" error-correction codes (ECCs) into this kind of optical codes is investigated and demonstrated with four families of ReedSolomon codes. With rich families of ECCs, our method enhances the collections of optical codes. The performances of these "treestructured" optical codes are algebraically analyzed and verified, for the first time, using the weight distribution of ECCs.Index Terms-Error-correction codes, optical code division multiple access (O-CDMA), Reed-Solomon (RS) codes, tree structure.

New Tree-Structured Optical Codes for Incoherent Optical-CDMA Applications

“…Because the optical codes in Section III and [13]- [16] are designed with each codeword using all available wavelengths at most once. Thus, codeword switching can be done all-optically by simply rearranging the wavelengths at different time slots, as illustrated in Fig.…”

“…In [15] and [16], a new relationship between non-binary ECCs [22]- [26] and 2-D optical codes was established. It was shown that distinct 2-D optical codewords could also be generated from the "translated" copies (i.e., repeatedly adding the all-"1" codeword) of ECCs, instead of cyclic-shifted copies [13], [14].…”

“…1. General relationship of cardinality and cross-correlation function among the levels and subsets of tree-structured optical codes [13]- [16], where λc is the maximum cross-correlation value of codewords within each subset, φ i is the cardinality of each subset in level i, m i is the number of subsets in level i that can be partitioned from each subset in level i + 1, and S = m n−1 m n−2 · · · m 2 , for i ∈ [1, n].…”

“…[13]- [16]. The large number of codewords in some of these codes can be partitioned into "tree" structures.…”

“…Section II studies how the aforementioned characteristics of these tree-structured 2-D optical codes [13]- [16] support contemporary applications, such as a scalable, multihop, multilevel O-CDMA network architecture with rapid code-switching/dropping [16]- [18] and enhancing physical-layer confidentiality in O-CDMA systems and networks by means of "multicode-keying" encryption [19]- [21]. Based on the maximal-distance-separable (MDS) codes [22]- [26], Section III shows and formally proves the constructions of two new families of 2-D optical codes.…”