The dimensions of Schur squares of HRS codes

Abstract

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 $p\ge t\ge 2s$ and $t\le \frac{r+2s-1}{2}$, the dimension of the Schur square of the HRS code $HR{S}_t(\{{\alpha }_1,\dots ,{\alpha }_r\},s)$ (with length $rs$ and dimension $t$) reaches the upper bound $(2t-2s+1)s$. In particular, when $p\ge t=2s$ and $r\ge t+1$, 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.