Grassmannian Frames in Composite Dimensions by Exponentiating Quadratic Forms

Grassmannian frames in composite dimensions D are constructed as a collection of orthogonal bases where each is the element-wise product of a mask sequence with a generalized Hadamard matrix. The set of mask sequences is obtained by exponentiation of a q-root of unity by different quadratic forms with m variables, where q and m are the product of the unique primes and total number of primes, respectively, in the prime decomposition of D. This method is a generalization of a well-known construction of mutually unbiased bases, as well as second-order Reed-Muller Grassmannian frames for power-of-two dimension D = 2 ^m , and allows to derive highly symmetric nested families of frames with finite alphabet. Explicit sets of symmetric matrices defining quadratic forms leading to constructions in non-prime-power dimension with good distance properties are identified.