Enumeration and Generation of $\rm PSL$ Equivalence Classes for Quad-Phase Codes of Odd Length

Abstract

Quad-phase codes of a given length can be grouped into equivalence classes based on operations preserving autocorrelation peak sidelobe level, or “ ${\rm PSL}$ -preserving operators.” The task of enumerating these equivalence classes is facilitated by establishing a relationship with the problem of enumerating equivalence classes of $2\times N$ binary grids with respect to a pair of binary grid symmetries. We show this connection by a one-to-one mapping between even-length quad-phase codes and $2\times N$ binary grids. When $N$ is even, this mapping allows the known enumeration of $2\times N$ 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-phase-code equivalence classes relative to the operations that preserve PSL. Lastly, we conduct exhaustive searches to determine the optimal ${\rm PSL}$ for quad-phase codes of lengths 25 and 26 (for the smaller of these lengths, the search space contains $4^{25}=2^{50}$ , 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 ${\rm PSL}$ 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 ${\rm PSL}$ is found to be $\sqrt{5}$ , achieved by $604\,480$ codes belonging to 9445 equivalence classes of size 64.