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arXiv · 2111.03684

Dense packings via lifts of codes to division rings

Nihar Gargava·Vlad Serban
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Abstract

We obtain algorithmically effective versions of the dense lattice sphere packings constructed from orders in $\mathbb{Q}$-division rings by the first author. The lattices in question are lifts of suitable codes from prime characteristic to orders $\mathcal{O}$ in $\mathbb{Q}$-division rings and we prove a Minkowski--Hlawka type result for such lifts. Exploiting the additional symmetries under finite subgroups of units in $\mathcal{O}$, we show this leads to effective constructions of lattices approaching the best known lower bounds on the packing density $\Delta_n$ in a variety of new dimensions $n$. This unifies and extends a number of previous constructions.

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Nihar Gargava, Vlad Serban. 2021-11-05. Dense packings via lifts of codes to division rings. https://arxiv.org/abs/2111.03684

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