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

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.