The Conway-Sloane calculus for 2-adic lattices

TL;DR

This paper explains and validates Conway and Sloane's 2-adic lattice calculus, a practical system for quadratic forms over 2-adic integers, enhancing computational efficiency over previous methods.

Contribution

It provides the first published proof of Conway and Sloane's 2-adic lattice calculus, confirming its correctness and utility.

Findings

Validated the Conway-Sloane system for 2-adic lattices

Demonstrated its superiority for calculations over earlier methods

Established a formal foundation for its use in quadratic form analysis

Abstract

We motivate and explain the system introduced by Conway and Sloane for working with quadratic forms over the 2-adic integers, and prove its validity. Their system is far better for actual calculations than earlier methods, and has been used for many years, but no proof has been published before now.