Title | Grassmannian Frames in Composite Dimensions by Exponentiating Quadratic Forms |

Publication Type | Conference Paper |

Year of Publication | 2020 |

Authors | Pitaval, R.-A., Qin, Y. |

Conference Name | 2020 IEEE International Symposium on Information Theory (ISIT) |

Date Published | jun |

Keywords | Delsarte-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 |

Abstract | 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 = 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. |

DOI | 10.1109/ISIT44484.2020.9174082 |

Citation Key | pitaval_grassmannian_2020 |