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arXiv · 1907.13068

High dimensional affine codes whose square has a designed minimum distance

Ignacio García-Marco·Irene Márquez-Corbella·Diego Ruano
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Abstract

Given a linear code $\mathcal{C}$, its square code $\mathcal{C}^{(2)}$ is the span of all component-wise products of two elements of $\mathcal{C}$. 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 $k(\mathcal{C})$ and high minimum distance of $\mathcal{C}^{(2)}$, $d(\mathcal{C}^{(2)})$? More precisely, given a designed minimum distance $d$ we compute an affine variety code $\mathcal{C}$ such that $d(\mathcal{C}^{(2)})\geq d$ and that the dimension of $\mathcal{C}$ is high. The best construction that we propose comes from hyperbolic codes when $d\ge q$ and from weighted Reed-Muller codes otherwise.

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Ignacio García-Marco, Irene Márquez-Corbella, Diego Ruano. 2019-07-30. High dimensional affine codes whose square has a designed minimum distance. https://arxiv.org/abs/1907.13068

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