Visible to the public Grassmannian Frames in Composite Dimensions by Exponentiating Quadratic Forms

TitleGrassmannian Frames in Composite Dimensions by Exponentiating Quadratic Forms
Publication TypeConference Paper
Year of Publication2020
AuthorsPitaval, R.-A., Qin, Y.
Conference Name2020 IEEE International Symposium on Information Theory (ISIT)
Date Publishedjun
KeywordsDelsarte-Goethals codes, different quadratic forms, element-wise product, explicit sets, exponentiation, generalized Hadamard matrix, Grassmannian frames, Hadamard matrices, Ker- dock codes, mask sequence, mutually unbiased bases, nonprime-power dimension, orthogonal bases, power-of-two dimension D, prime decomposition, pubcrawl, quadratic forms, Reed-Muller codes, Resiliency, Scalability, second-order Reed-Muller Grassmannian frames, set theory, unique primes
AbstractGrassmannian 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 = 2m, 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.
Citation Keypitaval_grassmannian_2020