Coherence Optimization and Best Complex Antipodal Spherical Codes

Vector sets with optimal coherence according to the Welch bound cannot exist for all pairs of dimension and cardinality. If such an optimal vector set exists, it is an equiangular tight frame and represents the solution to a Grassmannian line packing problem. Best Complex Antipodal Spherical Codes (BCASCs) are the best vector sets with respect to the coherence. By extending methods used to find best spherical codes in the real-valued Euclidean space, the proposed approach aims to find BCASCs, and thereby, a complex-valued vector set with minimal coherence. There are many applications demanding vector sets with low coherence. Examples are not limited to several techniques in wireless communication or to the field of compressed sensing. Within this contribution, existing analytical and numerical approaches for coherence optimization of complex-valued vector spaces are summarized and compared to the proposed approach. The numerically obtained coherence values improve previously reported results. The drawback of increased computational effort is addressed and a faster approximation is proposed which may be an alternative for time critical cases

A construction of strongly regular Cayley graphs and their applications to codebooks

In this paper, we give a kind of strongly regular Cayley graphs and a class of codebooks. Both constructions are based on choosing subsets of finite fields, and the main tools that we employed are Gauss sums. In particular, these obtained codebooks are asymptotically optimal with respect to the Welch bound and they have new parameters

Group Frames with Few Distinct Inner Products and Low Coherence

Frame theory has been a popular subject in the design of structured signals and codes in recent years, with applications ranging from the design of measurement matrices in compressive sensing, to spherical codes for data compression and data transmission, to spacetime codes for MIMO communications, and to measurement operators in quantum sensing. High-performance codes usually arise from designing frames whose elements have mutually low coherence. Building off the original "group frame" design of Slepian which has since been elaborated in the works of Vale and Waldron, we present several new frame constructions based on cyclic and generalized dihedral groups. Slepian's original construction was based on the premise that group structure allows one to reduce the number of distinct inner pairwise inner products in a frame with $n$ elements from $\frac{n(n-1)}{2}$ to $n-1$. All of our constructions further utilize the group structure to produce tight frames with even fewer distinct inner product values between the frame elements. When $n$ is prime, for example, we use cyclic groups to construct $m$-dimensional frame vectors with at most $\frac{n-1}{m}$ distinct inner products. We use this behavior to bound the coherence of our frames via arguments based on the frame potential, and derive even tighter bounds from combinatorial and algebraic arguments using the group structure alone. In certain cases, we recover well-known Welch bound achieving frames. In cases where the Welch bound has not been achieved, and is not known to be achievable, we obtain frames with close to Welch bound performance