High dimensional affine codes whose square has a designed minimum distance

07/30/2019

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by   Ignacio Garcia-Marco, et al.

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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.