arXiv subjects

Haojie Gu

Publications and source records attributed to Haojie Gu.

6 recordsLinked to original sources

arXiv2026

On Twisted Roth-Lempel Codes

In 1989, Roth and Lempel constructed a well-known family of non-Reed-Solomon maximum distance separable (MDS) codes. For decades, this family of codes has attracted extensive research attention due to its algebraic structure, low-complexity decoding, and broad applications in cryptography and data storage. In this paper, we present a class of twisted Roth-Lempel codes. We investigate their minimum distance, MDS and NMDS properties. Specifically, we determine the necessary and sufficient conditions for the TRL codes to have minimum distance n-k or n-k+1. Furthermore, we determine the necessary and sufficient conditions for the TRL code to be an MDS or NMDS code. Moreover, we show that the dimension of the Schur square of the TRL code is at least 2k+1, and thus the TRL code is a non-RS code inequivalent to the corresponding RL code.

cs.IT↗

arXiv2026

Two variants of Twisted Reed-Solomon Codes

Generalized Reed-Solomon codes and twisted generalized Reed-Solomon codes provide important sources of maximum distance separable codes. In this paper, we study two variants obtained by introducing column twists and simultaneous row-column twists into Reed-Solomon-type evaluation codes. For the column-twisted family, we provide necessary and sufficient conditions for the code to be MDS in terms of explicit subset product conditions. Under the stated parameter assumptions, the Schur square has dimension 2k+1, which leads to MDS codes that are not equivalent to Reed-Solomon codes. For the row-column twisted family, we establish necessary and sufficient conditions for the MDS property in terms of elementary symmetric functions. The larger Schur-square dimension provides a further distinction from both Reed-Solomon codes and known twisted families, thereby yielding new non-RS MDS codes. Finally, explicit parity-check matrices and dual descriptions are obtained for both code families. These results provide a foundation for subsequent studies of self-orthogonality, hull dimensions, and applications to quantum-code constructions.

cs.IT↗

arXiv2026

The dimensions of Schur squares of HRS codes

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.

cs.IT↗

arXiv2026

Unique Decoding of Hyperderivative Reed-Solomon Codes

Error-correcting codes are combinatorial objects designed to cope with the problem of reliable transmission of information on a noisy channel. A fundamental problem in coding theory and practice is to efficiently decode the received word with errors to obtain the transmitted codeword. In this paper, we consider the decoding problem of Hyperderivative Reed-Solomon (HRS) codes with respect to the NRT metric. Specifically, we propose a Welch-Berlekamp algorithm for the unique decoding of NRT HRS codes.

cs.IT↗

arXiv2025

Deep holes of a class of twisted Reed-Solomon codes

The deep hole problem is a fundamental problem in coding theory, and it has many important applications in code constructions and cryptography. The deep hole problem of Reed-Solomon codes has gained a lot of attention. As a generalization of Reed-Solomon codes, we investigate the problem of deep holes of a class of twisted Reed-Solomon codes in this paper. Firstly, we provide the necessary and sufficient conditions for $\boldsymbol{a}=(a_{0},a_{1},\cdots,a_{n-k-1})\in\mathbb{F}_{q}^{n-k}$ to be the syndrome of some deep hole of $TRS_{k}(\mathcal{A},l,η)$. Next, we consider the problem of determining all deep holes of the twisted Reed-Solomon codes $TRS_{k}(\mathbb{F}_{q}^{*},k-1,η)$. Specifically, we prove that there are no other deep holes of $TRS_{k}(\mathbb{F}_{q}^{*},k-1,η)$ for $\frac{3q+2\sqrt{q}-8}{4}\leq k\leq q-5$ when q is even, and $\frac{3q+3\sqrt{q}-5}{4}\leq k\leq q-5$ when q is odd. We also completely determine their deep holes for $q-4\leq k\leq q-2$ when $q$ is even.

cs.IT↗

arXiv2022

On twisted generalized Reed-Solomon codes with l twists

In this paper, we study a class of twisted generalized Reed-Solomon (TGRS) codes with general l twists. A sufficient and necessary condition for the TGRS codes to be MDS or l-MDS (l<k and l<n-k) is determined. A sufficient and necessary condition that such a TGRS code is self-dual for 3l<k-1 is also presented. Finally, we give an explicit construction of self-dual TGRS codes. And examples of self-dual MDS TGRS codes for small l are given.

cs.IT↗