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

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