Spectral Classification and Multiplicative Partitioning of Constant-Weight Sequences Based on Circulant Matrix Representation of Optical Orthogonal Codes

Considering the space of constant-weight sequences as the reference set for every optical orthogonal code (OOC) design algorithm, we propose a classification method that preserves the correlation properties of sequences. First, we introduce the circulant matrix representation of optical orthogonal codes and, based on the spectrum of circulant matrices, we define the spectral classification of the set S_n,w of all (0, 1)-sequences with length n, weight w, and the first chip “1”. Then, as a method for spectrally classifying the set S_n,w, we discuss an algebraic structure called multiplicative group action. Using the above multiplicative group action, we define an equivalence relation on S_n,w in order to classify it into equivalence classes called multiplicative partitions which are the same as the spectral classes. The algebraic properties of the proposed partitioning such as the number of classes and the size of each class are investigated and in the case of prime n, a novel formula for the number of classes is derived. Finally, we present and prove the autocorrelation, intraclass and interclass cross-correlation properties of our proposed classification of the space S_n,w that decrease the computational complexity of search algorithms in designing and constructing (n, w, λ_a,λ_c)-OOC.