The quadratic hull of a code and the geometric view on multiplication algorithms

TL;DR

This paper introduces the quadratic hull of a linear code and demonstrates its role in understanding symmetric bilinear multiplication algorithms for finite-dimensional algebras through a geometric lens, extending to evaluation at higher degree points.

Contribution

It defines the quadratic hull of a code and links it to multiplication algorithms via evaluation-interpolation, providing a geometric perspective and generalizing to multiplication reductions.

Findings

Quadratic hull characterizes multiplication algorithms geometrically.

Evaluation-interpolation at simple points on the quadratic hull yields all symmetric bilinear algorithms.

Examples include optimal algorithms for small algebras and connections to recent embedding concepts.

Abstract

We introduce the notion of quadratic hull of a linear code, and give some of its properties. We then show that any symmetric bilinear multiplication algorithm for a finite-dimensional algebra over a field can be obtained by evaluation-interpolation at simple points (i.e. of degree and multiplicity 1) on a naturally associated space, namely the quadratic hull of the corresponding code. This also provides a geometric answer to some questions such as: which linear maps actually are multiplication algorithms, or which codes come from supercodes (as asked by Shparlinski-Tsfasman-Vladut). We illustrate this with examples, in particular we describe the quadratic hull of all the optimal algorithms computed by Barbulescu-Detrey-Estibals-Zimmermann for small algebras. In our presentation we actually work with multiplication reductions. This is a generalization of multiplication algorithms, that allows for instance evaluation-interpolation at points of higher degree and/or with multiplicities, and also includes the recently introduced notion of "reverse multiplication-friendly embedding" from Cascudo-Cramer-Xing-Yang. All our results hold in this more general context.