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

In this work, we investigate a class of narrow-sense constacyclic BCH codes of length q2m-1a(q+1)$\frac{q^{2m}-1}{a(q+1)}$ over the finite field Fq2$\mathbb {F}_{q^2}$, where q is a prime power, m≥2$m\ge 2$ is an even integer, and a≠1$a\ne 1$ is a divisor of q-1$q-1$. The maximum designed distances such that narrow-sense constacyclic BCH codes contain their Hermitian dual codes are determined. The dimensions of the corresponding Hermitian dual-containing codes are worked out. Further, the related quantum codes are constructed. The construction improves the parameters of quantum codes available in the literature.

In this paper, we first give two new methods for constructing self-orthogonal codes from known self-orthogonal codes. On the basis of this, we construct four infinite classes of binary self-orthogonal codes. Moreover, we also determine their weight distributions and the minimum distances of their dual codes. Furthermore, we present a class of optimal linear codes and a class of almost optimal linear codes with respect to the Sphere Packing Bound.

In this work, we study and determine the dimensions of Euclidean and Hermitian hulls of two classical propagation rules, namely, the (u,u+v) -construction and the direct sum construction. Some new criteria for the resulting codes derived from these two propagation rules being self-dual, self-orthogonal, or linear complementary dual (LCD) codes are given. As applications, we employ the (u,u+v) -construction to obtain (almost) self-orthogonal codes; employ the direct sum construction to provide lower bounds on the minimum distance of FSD (LCD) codes; and employ both these two constructions to derive linear codes with prescribed hull dimensions. Many (almost) optimal codes are presented. In particular, a family of binary almost Euclidean self-orthogonal Griesmer codes is constructed. We also obtain many binary, ternary Euclidean and quaternary Hermitian FSD LCD codes of larger lengths and improve some lower bounds on the minimum distance of known ternary Euclidean LCD codes.

Galois self-orthogonal (SO) codes are generalizations of Euclidean and Hermitian SO codes. Algebraic geometry (AG) codes are the first known class of linear codes exceeding the Gilbert-Varshamov bound. Both of them have attracted much attention for their rich algebraic structures and wide applications in these years. In this paper, we consider them together and study Galois SO AG codes. A criterion for an AG code being Galois SO is presented. Based on this criterion, we construct several new classes of maximum distance separable (MDS) Galois SO AG codes from projective lines and several new classes of Galois SO AG codes from projective elliptic curves, hyper-elliptic curves and hermitian curves. In addition, we give an embedding method that allows us to obtain more MDS Galois SO codes from known MDS Galois SO AG codes.

Quantum secret sharing is one of the important techniques in quantum cryptography. In this paper, we propose a novel dynamic hierarchical quantum secret sharing scheme with general access structure. Participants from different levels share the same secret. Firstly, a special hierarchical structure based on the generalized GHZ state is constructed, which expands the application value of the existing hierarchical quantum secret sharing. Secondly, this paper uses the monotone span program (MSP) and the generalized Pauli operator to realize the dynamic property of the scheme, which includes three aspects: The hierarchical access structure is variable; participants can join or leave, and the shared secret can be updated. Moreover, the shares of the participants can be protected so as to reduce communication consumption due to reuse of the shares. Finally, compared with other hierarchical quantum secret sharing schemes, the proposed scheme is not only more flexible but also more secure.

Hulls of linear codes are widely studied due to their good properties and wide applications. Let $n=\frac{q^m-1}{r}$ and $\mathcal {C}$ be an [n, k] cyclic code over $\mathbb {F}_q$, where $r|q-1$. In this paper, we present several necessary and sufficient conditions for BCH codes of length n that have $k-1$ or $k^\perp -1$ dimensional hulls, where $k^\perp$ is the dimension of $\mathcal {C}^\perp$. Further, we give the parameters of several families of self-orthogonal codes that arise as hulls of BCH codes. We obtain many optimal codes.

The object of this paper is to study two very important classes of codes in coding theory, namely self-orthogonal (SO) and linear complementary dual (LCD) codes under the symplectic inner product, involving characterization, constructions, and their application. Using such a characterization, we determine the mass formulas of symplectic SO and LCD codes by considering the action of the symplectic group, and further obtain some asymptotic results. Finally, under the Hamming distance, we obtain some symplectic SO (resp. LCD) codes with improved parameters directly compared with Euclidean SO (resp. LCD) codes. Under the symplectic distance, we obtain some additive SO (resp. additive complementary dual) codes with improved parameters directly compared with Hermitian SO (resp. LCD) codes. Further, we also construct many good additive codes outperform the best-known linear codes in Grassl's code table. As an application, we construct a number of record-breaking (entanglement-assisted) quantum error-correcting codes, which improve Grassl's code table.

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.