High dimensional affine codes whose square has a designed minimum distance

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 the dimension of C is high. The best constructions we propose mostly come from hyperbolic codes. Nevertheless, for small values of d, they come from weighted Reed-Muller codes.