New lattice sphere packings denser than Mordell-Weil lattices

TL;DR

This paper introduces new dense lattice sphere packings in high dimensions, surpassing known records and Mordell-Weil lattices, through novel constructions based on analogues of Craig lattices and potential binary code applications.

Contribution

It presents the first known lattice sphere packings denser than Mordell-Weil lattices in many high dimensions and introduces new dense lattices in moderate dimensions using Craig lattice analogues.

Findings

New dense lattice sphere packings in dimensions 3332-4096 surpass Mordell-Weil lattices.

Construction of lattices with densities at least 8 times that of Craig lattices in certain prime-related dimensions.

Record-breaking lattice sphere packings in dimensions 4098-8232 and moderate dimensions 84-189.

Abstract

1) We present new lattice sphere packings in Euclid spaces of many dimensions in the range 3332-4096, which are denser than known densest Mrodell-Weil lattice sphere packings in these dimensions. Moreover it is proved that if there were some nice linear binary codes we could construct lattices denser than Mordell-Weil lattices of many dimensions in the range 116-3332. 2) New lattices with densities at least 8 times of the densities of Craig lattices in the dimensions $p-1$, where $p$ is a prime satisfying $p-1 \geq 1222$, are constructed. Some of these lattices provide new record sphere packings. 3) Lattice sphere packings in many dimensions in the range 4098-8232 better than present records are presented. Some new dense lattice sphere packings in moderate dimensions $84, 85, 86, 181-189$ denser than any previously known sphere packings in these dimensions are also given. The construction is based on the analogues of Craig lattices.