Infinite families of optimal systems of biangular lines related to representations of SL(2,Fq)

Mikhail Ganzhinov

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    Abstract

    A line packing is optimal if its coherence is as small as possible. Most interesting examples of optimal line packings are achieving equality in some of the known lower bounds for coherence. In this paper two infinite families of real and complex biangular line packings are presented. New packings achieve equality in the real or complex second Levenshtein bound respectively. Both infinite families are constructed by analyzing well known representations of the finite groups SL(2,Fq). Until now the only known infinite families meeting the second Levenshtein bounds were related to the maximal sets of mutually unbiased bases (MUB). Similarly to the line packings related to the maximal sets of MUBs, the line packings presented here are related to the maximal sets of mutually unbiased weighing matrices. Another similarity is that the new packings are projective 2-designs. The latter property together with sufficiently large cardinalities of the new packings implies some improvement on largest known cardinalities of real and complex biangular tight frames.

    Original languageEnglish
    Article number105656
    Number of pages21
    JournalJournal of Combinatorial Theory. Series A
    Volume192
    DOIs
    Publication statusPublished - Nov 2022
    MoE publication typeA1 Journal article-refereed

    Keywords

    • Combinatorics
    • Functional analysis
    • Levenshtein's bounds
    • Line packings

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