Enumeration and Generation of PSL Equivalence Classes for Quad-Phase Codes of Even Length

Abstract

Quad-phase codes of a given length can be grouped into equivalence classes based on operations preserving autocorrelation peak sidelobe level, or “PSL-preserving operators.” The task of enumerating these equivalence classes is facilitated by establishing a relationship with the problem of enumerating equivalence classes of 2 x N binary grids with respect to a pair of binary grid symmetries. We show this connection by a one-to-one mapping between even-length quadphase codes and 2 x N binary grids. When N is even, this mapping allows the known enumeration of 2 x N binary grids to be applied to the enumeration of even-length quad-phase code PSL equivalence classes. Furthermore, this equivalence instructs the development of a simpler algorithm for visiting single representatives of quad-phasecode equivalence classes relative to the operations that preserve PSL. Lastly, we conduct exhaustive searches to determine the optimal PSL for quad-phase codes of lengths 25 and 26 (for the smaller of these lengths, the search space contains 4 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">25</sup> = 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">50</sup> , or roughly 1 quadrillion codes). One of the devices used to narrow the search space while achieving an exhaustive search is to seed the search with the equivalence class representatives of length N = 6. The optimal PSL for length-25 quad-phase codes is determined to be 2, achieved by the codes in a single size-32 equivalence class. For length 26, the optimal PSL is found to be √5, achieved by 604480 codes belonging to 9445 equivalence classes of size 64.