On the geometry of nearly orthogonal lattices

Fukshansky, Lenny; Kogan, David

doi:10.1016/j.laa.2021.07.015

Cited by 3 publications

(2 citation statements)

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“…Hence it cannot be smaller than p, and so (7) guarantees that C(Λ K ) ≤ 1 p−1 . Thus we have the result.…”

Section: Coherence Of Cyclotomic Latticesmentioning

confidence: 99%

“…These values do not depend on the specific choice of vectors in S (L) out of each ± pair. If L has a basis of minimal vectors, then A(L) becomes a certain alternative measure of its "non-orthogonality": A(L) ≥ 0 with equality if and only if S (L) is an orthogonal basis for L. Maximal coherence on lattices has previously been introduced in [8] and studied on nearly orthogonal lattices in [7], but average coherence has not previously been extended to lattices, as far as we know. Average coherence for frames was introduced in [2].…”

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confidence: 99%

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On average coherence of cyclotomic lattices

Fukshansky¹,

Kogan²

2022

Communications in Mathematics

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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.

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“…Hence it cannot be smaller than p, and so (7) guarantees that C(Λ K ) ≤ 1 p−1 . Thus we have the result.…”

Section: Coherence Of Cyclotomic Latticesmentioning

confidence: 99%

mentioning

confidence: 99%