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.

中文翻译：

---

正方形具有设计的最小距离的高维仿射码

给定线性代码 $\mathcal{C}$，其平方代码 $\mathcal{C}^{(2)}$ 是 $\mathcal{C}$ 的两个元素的所有分量乘积的跨度。受多方计算应用的启发，我们开展这项工作的目的是回答以下问题：哪些仿射变异码家族同时具有高维数 $k(\mathcal{C})$ 和 $\mathcal{ 的高最小距离C}^{(2)}$, $d(\mathcal{C}^{(2)})$? 更准确地说，给定设计的最小距离 $d$，我们计算仿射变异代码 $\mathcal{C}$，使得 $d(\mathcal{C}^{(2)})\geq d$ 以及$\mathcal{C}$ 很高。我们建议的最佳构造来自双曲码，当 $d\ge q$ 时，来自加权 Reed-Muller 码。