Shixin Zhu’s research while affiliated with Hefei University of Technology and other places

Classical locally recoverable codes (classical LRCs) have wide application in large-scale distributed storage systems due to their efficiency in the recovery of localized errors. In order to satisfy the quest for quantum data storage, quantum locally recoverable codes (qLRCs) have been proposed for their relevant and prospective application. Firstly, two families of classical LRCs from subfields and cyclic groups of finite fields are constructed and their duals are proved to be dual-containing. Then two families of optimal qLRCs with regard to the quantum Singleton-like bound on qLRCs are deduced.

Galois hulls of linear codes are generalizations of the Euclidean and Hermitian hulls of linear codes. Propagation rules are efficient ways to derive new linear codes from known initial linear codes. In this paper, we combine these two topics and obtain some general results on the dimensions of Galois hulls of linear codes derived from an improved propagation rule. As applications, we further derive many optimal or almost optimal (almost) Galois self-orthogonal codes by employing Galois self-orthogonal maximum distance separable codes and punctured simplex type codes as initial linear codes. In particular, two families of almost Euclidean self-orthogonal binary (near) Griesmer codes are obtained.

In this paper, we employ group rings and skew group rings to construct binary self-dual codes of length 72. There is a well known map $\sigma$ that sends a (skew) group ring element to an $n\times n$ matrix. By using groups of order n, we obtain a lot of generator matrices of the form $(I_{n}\mid \sigma _{\varphi }(v)$), where v is an element in a (skew) group ring. Then through special maps, we can send a self-dual code of length 2n over ${\mathbb {F}}_{2^t}$ to a binary self-dual code of length 2tn. We use these generator matrices to search for binary [72,36,12] self-dual codes and obtain many singly-even and doubly-even codes with new parameters in their weight enumerators that were not known in the literature before. We list our findings on a publicly available website.

Maximum distance separable (MDS) codes that are not monomially equivalent to generalized Reed-Solomon (GRS) codes are called non-GRS MDS codes, which have important applications in communication and cryptography. Covering radii and deep holes of linear codes are closely related to their decoding problems. In the literature, the covering radii and deep holes of GRS codes have been extensively studied, while little is known about non-GRS MDS codes. In this paper, we study two classes of extended twisted generalized Reed-Solomon (ETGRS) codes involving their non-GRS MDS properties, covering radii, and deep holes. In other words, we obtain two classes of non-GRS MDS codes with known covering radii and deep holes. As applications, we further directly derive more non-GRS MDS codes, and get some results on the existence of their error-correcting pairs. As a byproduct, we find some connections between the well-known Roth-Lempel codes and these two classes ETGRS codes.

In this paper, we employ group rings and automorphism groups of binary linear codes to construct new record-breaking binary linear codes. We consider the semidirect product of abelian groups and cyclic groups and use these groups to construct linear codes. Finally, we obtain some linear codes which have better parameters than the code in \cite{bib5}. All the calculation results and corresponding data are listed in the paper or posted online.

In this paper, a class of narrow-sense constacyclic BCH codes over $\mathbb {F}_{q^2}$ with length $n=\frac{q^{2m}-1}{2\left( q^2-1\right) }$ is studied, where $q\ge 3$ is an odd prime power and $m\ge 2$ is even. The maximum designed distance such that narrow-sense constacyclic BCH codes over $\mathbb {F}_{q^2}$ with length n containing their Hermitian dual codes is determined. We obtain some new quantum codes by using such narrow-sense constacyclic BCH codes. Our constructions not only have larger designed distance but also have better parameters than the ones in the literature.

Entanglement-assisted quantum error-correcting codes (EAQECCs) not only can boost the performance of stabilizer quantum error-correcting codes but also can be derived from arbitrary classical linear codes by loosing the self-orthogonal condition and using pre-shared entangled states between the sender and the receiver. It is a challenging work to construct optimal EAQECCs and determine the required number of pre-shared entangled states. Let $\mathcal {R}_{t}=\mathbb {F}_{q^{2}}+v\mathbb {F}_{q^{2}}+v^{2}\mathbb {F}_{q^{2}}+\cdots +v^{t}\mathbb {F}_{q^{2}}$, where q is an odd prime power and $v^{t+1}=1$. Based on the generalized Gray map that is provided from $\mathcal {R}_{t}$ to $\mathbb {F}_{q^{2}}^{t+1}$, some new optimal EAQECCs are constructed from the Gray images of v-constacyclic codes over $\mathcal {R}_{t}$. Compared with the known ones, our codes have better parameters.

Let $q=p^m$ be a prime power, e be an integer with $0\leq e\leq m-1$ and $s=\gcd(e,m)$. In this paper, for a vector v and a q-ary linear code C, we give some necessary and sufficient conditions for the equivalent code vC of C and the extended code of vC to be e-Galois self-orthogonal. From this, we directly obtain some necessary and sufficient conditions for (extended) generalized Reed-Solomon (GRS and EGRS) codes to be e-Galois self-orthogonal. Furthermore, for all possible e satisfying $0\leq e\leq m-1$, we classify them into three cases (1) $\frac{m}{s}$ odd and p even; (2) $\frac{m}{s}$ odd and p odd; (3) $\frac{m}{s}$ even, and construct several new classes of e-Galois self-orthogonal maximum distance separable (MDS) codes. It is worth noting that our e-Galois self-orthogonal MDS codes can have dimensions greater than $\lfloor \frac{n+p^e-1}{p^e+1}\rfloor$, which are not covered by previously known ones. Moreover, by propagation rules, we obtain some new MDS codes with Galois hulls of arbitrary dimensions. As an application, many quantum codes can be obtained from these MDS codes with Galois hulls.