|Grassmannian Frames in Composite Dimensions by Exponentiating Quadratic Forms
|Renaud-Alexandre Pitaval, Yi Qin, Huawei Technologies Sweden AB, Sweden
|A.1: Algebraic Coding Theory I
|Algebraic and Combinatorial Coding Theory
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|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.