High dimensional affine codes whose square has a designed minimum distance

Abstract

Given a linear code C, its square code C (2) is the span of all component-wise products of two elements of C. Motivated by applications in multi-party computation, our purpose with this work is to answer the following question: which families of affine variety codes have simultaneously high dimension k(C) and high minimum distance of C (2) , d(C (2) )? More precisely, given a designed minimum distance d we compute an affine variety code C such that d(C (2) ) ≥ d and that the dimension of C is high. The best construction that we propose comes from hyperbolic codes when d ≥ q and from weighted Reed-Muller codes otherwise.