High dimensional affine codes whose square has a designed minimum distance

TL;DR

This paper investigates affine variety codes with high dimension and a square code that maintains a high minimum distance, crucial for multi-party computation, proposing constructions from hyperbolic and weighted Reed-Muller codes.

Contribution

It introduces new constructions of affine variety codes ensuring high dimension and minimum distance of the square code, advancing code design for cryptographic applications.

Findings

Hyperbolic codes achieve high minimum distance when d ≥ q.

Weighted Reed-Muller codes are effective for smaller d.

Constructed codes meet the designed minimum distance criteria.

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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