A new class of near-optimal partial Fourier codebooks from an almost difference set

An (N, K) codebook is a set of N unit-norm code vectors in a K-dimensional vector space. Also known as a frame, it has many applications in communications, signal processing, and quantum computing. In the applications, it is required that the maximum magnitude of inner products between a pair of distinct code vectors should meet the Welch bound equality, strictly or asymptotically. In this paper, a new class of (N, K) partial Fourier codebooks is constructed from an almost difference set, where N = K2 − 1 and K = pk for a prime p and a positive integer k. It turns out that the almost difference set is equivalent to a modular Golomb ruler, and is obtained by a set of elements decimated from an N-ary Sidelnikov sequence of length N with decimation factor K − 1. In the codebook, the magnitude of inner products between distinct code vectors is two-valued, and its maximum nearly achieves the Welch bound equality, which leads to a near-optimal codebook or nearly equiangular tight frame. Equivalent to a K × N partial Fourier matrix with near-optimal coherence, the new partial Fourier codebook can find its potential applications in deterministic compressed sensing.