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Dense Packings via Lifts of Codes to Division Rings by Nihar Gargava; Vlad Serban is a Computer Science article available to read on EtoBox.

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Horlemann-Trautmann, Associate Editor for Coding and Decoding. Digital Object Identifier 10.1109/TIT.2022.3226028 increases and with giving effective constructions of lattices approaching these bounds. The classical Minkowski-Hlawka theorem [7] states that Δ n ≥ 2 ζ(n) 2 n , a bound which Rogers [8] later improved by a linear factor to Δ n ≥ cn 2 n for c = 2/e ≈ 0.74. The constant was subsequently sharpened to c = 1.68 by Davenport-Rogers [9] and c = 2 by Ball [10] for all n. More recently, Vance [11] showed using lattices which are modules over the Hurwitz integers that one may take c = 24/e ≈ 8.83 and Venkatesh [12] showed that for n large enough one may take c = 65963. Moreover, by considering lattices from maximal orders in cyclotomic fields, Venkatesh was able to achieve for infinitely many dimensions the improvement Δ n ≥ n log log n 2 n+1 . The first author [13] then extended such results to lattices coming from orders O in arbitrary Q-division algebras. This was achieved by proving a Siegel mean value theorem (see [13], [14]) in this setting and exploiting the additional symmetries of the lattices under the group of finite order units in O × to obtain dense packings. In par

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Author

Nihar Gargava; Vlad Serban

Publisher

Institute of Electrical and Electronics Engineers (IEEE)

Published

2023

Language

EN

Field

Computer Science (Physical Sciences)