Error-correcting codes over the Mordell-Weil groups of extremal rational elliptic surfaces and the $E_8$ lattice

TL;DR

This paper constructs the $E_8$ lattice using error-correcting codes derived from the Mordell-Weil groups of rational elliptic surfaces with maximal singularity lattice rank, extending classical code lattice methods.

Contribution

It introduces a novel construction of the $E_8$ lattice from codes over Mordell-Weil groups, generalizing known code lattice constructions like Construction A.

Findings

Constructed the $E_8$ lattice from Mordell-Weil group codes.

Connected lattice construction to Lie algebraic extensions.

Extended classical code lattice methods to elliptic surface context.

Abstract

We construct the $E_8$ lattice from classical error-correcting codes over the Mordell-Weil groups of rational elliptic surfaces that have a singularity lattice of rank 8 (maximal) for all cases of Oguiso-Shioda's classification. By the structure theorem of the Mordell-Weil lattice of rational elliptic surfaces, if the rank of the singularity lattice is maximal, then the Mordell-Weil group is a cyclic group or a direct sum of them. The singularity lattices are glued together by a code over their natural ring to form the $E_8$ lattice. Such constructions of the $E_8$ lattice from codes can be seen as a Lie algebraic extension and further generalization of known code lattice constructions such as Construction A and Construction A${}_{\rm C}$.