The Schur square of linear codes over a finite field has emerged as a fundamental operation in both classical and quantum coding theory. In this paper, we investigate the Schur square problem of Hyperderivative Reed-Solomon (HRS) codes. By solving certain special determinants, we first give a lower bound and an upper bound for the dimensions of Schur squares of HRS codes, and then prove that when $p\geq t\geq 2s$ and $t\leq \frac{r+2s-1}{2}$, the dimension of the Schur square of the HRS code $HRS_{t}(\{α_{1},\dots,α_{r}\},s)$ (with length $rs$ and dimension $t$) reaches the upper bound $(2t-2s+1)s$. In particular, when $p \ge t=2s$ and $r\geq t+1$, the dimension of the Schur square equals $\frac{t(t+1)}{2}$ which is the dimension of the Schur squares of random codes with high probability. As an application in code-based cryptography, HRS codes with specific parameter settings might resist the attack of Schur square distinguisher.

翻译：有限域上线性码的Schur平方已成为经典编码理论与量子编码理论中的基本运算。本文研究超导里德-所罗门(HRS)码的Schur平方问题。通过求解特定行列式，我们首先给出HRS码Schur平方维数的下界与上界，进而证明当$p\geq t\geq 2s$且$t\leq \frac{r+2s-1}{2}$时，HRS码$HRS_{t}(\{α_{1},\dots,α_{r}\},s)$（长为$rs$，维数为$t$）的Schur平方维数达到上界$(2t-2s+1)s$。特别地，当$p \ge t=2s$且$r\geq t+1$时，其Schur平方维数等于$\frac{t(t+1)}{2}$，与随机码的Schur平方维数以高概率取得的数值相同。作为基于编码密码学中的一项应用，特定参数设置的HRS码可能抵御Schur平方区分攻击。