Ruhao Wan’s research while affiliated with Hefei University of Technology and other places

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.

Quantum maximum-distance-separable (MDS) codes are an important class of quantum codes. In recent years, Hermitian self-orthogonal generalized Reed-Solomon (GRS) codes have been widely used to construct quantum MDS codes. In this paper, we give some sufficient conditions under which a certain system of equations over ${\mathbb {F}}_{q^2}$ has a solution over ${\mathbb {F}}_q^*$, which effectively unify similar known techniques for constructing Hermitian self-orthogonal codes. Moreover, we construct five new classes of q-ary quantum MDS codes with flexible parameters from Hermitian self-orthogonal GRS codes. Compared to the previous literature, the quantum MDS codes we construct have different lengths, or the same length but larger distances. In particular, some of the quantum MDS codes we construct have distances that can be taken to the maximum distance of the quantum MDS codes from GRS codes.

MDS self-dual codes have nice algebraic structures and are uniquely determined by lengths. Recently, the construction of MDS self-dual codes of new lengths has become an important and hot issue in coding theory. In this paper, we construct six new classes of MDS self-dual codes by using generalized Reed-Solomon (GRS for short) codes and extended GRS codes. Together with our constructions, the proportion of all known MDS self-dual codes relative to possible MDS self-dual codes generally exceed 57%. As far as we know, this is the largest known ratio. Moreover, some new families of MDS self-orthogonal codes are also constructed.

The hull of a linear code is the intersection of itself with its dual code with respect to certain inner product. Both Euclidean and Hermitian hulls are of theorical and practical significance. In this paper, we construct several new classes of maximum distance separable (MDS) codes via (extended) generalized Reed-Solomon (GRS) codes and determine their Euclidean or Hermitian hulls. As a consequence, four new classes of MDS codes with Hermitian hulls of flexible dimensions and six new classes of MDS codes with Euclidean hulls of flexible dimensions are constructed. As applications, for the former, we further construct four new families of entanglement-assisted quantum error-correcting codes (EAQECCs) and four new families of MDS EAQECCs of length n>q+1$n>q+1$. Meanwhile, many of the distance parameters of our MDS EAQECCs are greater than ⌈q2⌉$\lceil \frac{q}{2} \rceil$ or q; for the latter, we show some examples on Euclidean self-orthogonal and one-dimensional Euclidean hull MDS codes. In addition, two new general methods for constructing extended GRS codes with (k-1)$(k-1)$-dimensional Hermitian hull and Hermitian self-orthogonal extended GRS codes are also provided.

It is important task to construct quantum maximum-distance-separable (MDS for short) codes with good parameters. In this paper, by using Hermitian self-orthogonal generalized Reed-Solomon (GRS for short) codes, we construct four new classes of quantum MDS codes. Our quantum MDS codes have flexible parameters. And the minimum distances of our quantum MDS codes can be larger than q/2+1. Furthermore, it turns out that our constructions generalize and improve some previous results.

MDS self-dual codes have nice algebraic structures, theoretical significance and practical implications. In this paper, we present three classes of $q^2$-ary Hermitian self-dual (extended) generalized Reed-Solomon codes with different code locators. Combining the results in Ball et al. (Designs, Codes and Cryptography, 89: 811-821, 2021), we show that if the code locators do not contain zero, $q^2$-ary Hermitian self-dual (extended) GRS codes of length $\geq 2q\ (q>2)$ does not exist. Under certain conditions, we prove Conjecture 3.7 and Conjecture 3.13 proposed by Guo and Li et al. (IEEE Communications Letters, 25(4): 1062-1065, 2021).

MDS self-dual codes have nice algebraic structures and are uniquely determined by lengths. Recently, the construction of MDS self-dual codes of new lengths has become an important and hot issue in coding theory. In this paper, we develop the existing theory and construct six new classes of MDS self-dual codes. Together with our constructions, the proportion of all known MDS self-dual codes relative to possible MDS self-dual codes generally exceed 57\%. As far as we know, this is the largest known ratio. Moreover, some new families of MDS self-orthogonal codes and MDS almost self-dual codes are also constructed.

In this paper, we construct some new classes of maximum distance separable (MDS) self-dual codes over $F_q$ by using (extended) generalized Reed-Solomon codes. In this paper, for finite field with odd characteristic, we construct some new classes of MDS self-dual codes than previous works.