# 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.

## Key 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.