Lenny Fukshansky ; David Kogan - On average coherence of cyclotomic lattices

cm:10241 - Communications in Mathematics, November 7, 2022, Volume 31 (2023), Issue 1 - https://doi.org/10.46298/cm.10241
On average coherence of cyclotomic lattices

Authors: Lenny Fukshansky ; David Kogan

    We introduce maximal and average coherence on lattices by analogy with these notions on frames in Euclidean spaces. Lattices with low coherence can be of interest in signal processing, whereas lattices with high orthogonality defect are of interest in sphere packing problems. As such, coherence and orthogonality defect are different measures of the extent to which a lattice fails to be orthogonal, and maximizing their quotient (normalized for the number of minimal vectors with respect to dimension) gives lattices with particularly good optimization properties. While orthogonality defect is a fairly classical and well-studied notion on various families of lattices, coherence is not. We investigate coherence properties of a nice family of algebraic lattices coming from rings of integers in cyclotomic number fields, proving a simple formula for their average coherence. We look at some examples of such lattices and compare their coherence properties to those of the standard root lattices.


    Volume: Volume 31 (2023), Issue 1
    Published on: November 7, 2022
    Accepted on: November 2, 2022
    Submitted on: November 1, 2022
    Keywords: Mathematics - Number Theory,11H06, 11H31, 11R18, 42C15

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