arXiv subjects

Xiuling Shan

Publications and source records attributed to Xiuling Shan.

4 recordsLinked to original sources

arXiv2026

Geometric construction of k-optimal locally repairable codes

A linear code is referred to as a locally repairable code (LRC) with locality r if any erased code symbol can be recovered by accessing at most r other code symbols. LRCs are highly desirable for distributed storage systems to enhance repair efficiency. In this paper, we investigate LRCs with disjoint repair sets via the parity-check matrix method. Firstly, we propose a novel concept of the s-Pasch configuration and present a geometric characterization for the existence of LRCs with minimum distance 5 and locality 3. Subsequently, we construct k-optimal LRCs by exploiting the point-line relationship in PG(2,q). Finally, a family of q-ary k-optimal LRCs with minimum distance 6 and general locality r is constructed using partial r-spreads.

cs.IT↗

arXiv2026

Constructions of two-dimensional optical orthogonal codes of weight three

The study of optical orthogonal codes has been motivated by an application in an optical code-division multiple access system. This paper focuses on optimal two-dimensional optical orthogonal codes with autocorrelation and cross-correlation both equal to $1$. By examining the structures of $n$-cyclic group divisible packings and semi-cyclic incomplete holey group divisible designs, we present new combinatorial constructions for two-dimensional $(m\times n,k,1)$-optical orthogonal codes. As a consequence, the exact number of codewords of an optimal two-dimensional $(m\times n,3,1)$-optical orthogonal code is determined for any positive integers $m$ and $n$.

math.CO↗

arXiv2025

On the cardinalities of quantum Latin squares

A quantum Latin square of order $v$, QLS($v$), is a $v\times v$ array in which each of entries is a unit column vector from the Hilbert space $\mathbb{C}^{v}$, such that every row and column forms an orthonormal basis of $\mathbb{C}^{v}$. The cardinality of a QLS($v$) is the number of its vectors distinct up to a global phase, which is the crucial indicator for distinguishing between classical QLSs and non-classical QLSs. In this paper, we investigate the possible cardinalities of a QLS($v$). As a result, we completely resolve the existence of a QLS($v$) with maximal cardinality for any $v\geq 4$. Moreover, based on Wilson's construction and Direct Product construction, we establish some possible cardinality range of a QLS($v$) for any $v\geq 4$.

math.CO↗

arXiv2022

A new chaotic image encryption algorithm based on transversals in a Latin square

There are some good combinatorial structures suitable for image encryption. In this study, a new chaotic image encryption algorithm based on transversals in a Latin square is proposed. By means of an n-transversal of a Latin square, we can permutate an image data group by group for the first time, then use two Latin squares for auxiliary diffusion on the basis of a chaotic sequence, finally make use of a pair of orthogonal Latin squares to do the second scrambling. As a whole, the encryption process of "scrambling-diffusion-scrambling" is formed. The experimental results indicate that this algorithm achieves a secure and fast encryption effect. The final information entropy is very close to 8, and the correlation coefficient is approximate to 0. All these tests verify the robustness and practicability of this proposed algorithm.

cs.CR↗