arXiv subjects

San Ling

Publications and source records attributed to San Ling.

At least 19 recordsLinked to original sources

arXiv2026

Plotkin-optimal pure quantum LRCs with large minimum distances from NMDS codes

Quantum locally recoverable codes (qLRCs) are a class of quantum error-correcting codes that can recover an erased qudit by accessing only a few other qudits. This paper develops a general framework for constructing optimal pure qLRCs that attain the Plotkin-like bound, referred to as Plotkin-optimal pure qLRCs, from Hermitian dual-containing near maximum-distance separable (NMDS) codes. Focusing on a special class of NMDS codes called extended subcodes of generalized Reed-Solomon (ESGRS) codes, we determine their localities by analyzing certain subset sum problems over finite fields. By further exploiting the algebraic structures of ESGRS codes and related codes, we construct four families of Plotkin-optimal pure qLRCs with large lengths and more flexible minimum distances than existing constructions. These constructions provide strong global error-correction capability while retaining local recovery. To the best of our knowledge, they give the first families of qLRCs attaining the pure Plotkin-like bound. In addition, by using the resulting locality property, we resolve in the negative a related problem on ESGRS codes supporting combinatorial designs.

cs.IT↗

arXiv2026

Proofs and disproofs of some conjectures on zero-sum problems of new types

For integers $r\ge1$ and $n\ge2$, let $s_r(n)$ be the least $N$ such that every sequence of $N$ vectors in $\Z^r$, each nonzero modulo $n$, contains $n$ terms whose sum is zero modulo $n$ but nonzero modulo $n^2$. Let $w_r(n)$ be the corresponding least $N$ when any nonempty subsequence is allowed. These constants were introduced as zero-sum problems of new types by Z.-W. Sun. We prove $s_r(n)\ge(2^{r+1}-2)(n-1)+1$ by constructing an explicit sequence from a lattice polytope, \emph{disproving} his conjecture $s_2(n)=4n+1$ for $n\ge4$. We obtain $w_r(2)=2r+1$ by classifying all sequences avoiding such nonempty subsequences modulo $4$. We also \emph{prove} Sun's conjecture $w_2(n)=4n-3$ for $n=2^k3^\ell p$, where $p$ is a prime and $k,\ell\ge0$ are integers, by classifying the extremal sequences of length $4p-4$ modulo $p^2$ when $n=p$. For every fixed $n\ge2$, we prove $w_r(n)/s(\Z_n^r)=O_n(r^2/2^r)\to0$ as $r\to\infty$, where $s(\Z_n^r)$ is the classical Erdős-Ginzburg-Ziv constant, with an explicit threshold on $r$ for strict inequality. Thus, for every fixed modulus $n\ge2$, Sun's proposed equality fails for sufficiently large $r$.

math.CO↗

arXiv2026

Embedding linear codes over Z4 into self-orthogonal codes

The purpose of this paper is to investigate the self-orthogonal embedding problem for linear codes over Z4. We propose several tight bounds on the length of the shortest self-orthogonal embedding over Z4, and determine the exact shortest self-orthogonal embedding length under specific conditions. As an example satisfying these conditions, we establish the exact length of the shortest self-orthogonal embedding for the quaternary Preparata codes. Furthermore, to establish these results, we completely classify the exact length of the shortest doubly even self-orthogonal embedding for binary linear codes in every possible case. Finally, when the shortest self-orthogonal embedding length of a given free code over Z4 is equal to the shortest doubly even self-orthogonal embedding length of its residue code, we present an algorithm to construct all possible shortest self-orthogonal embeddings. With our algorithm, we found twelve linear codes over Z4 whose minimum Lee distances are higher than those of the Z4-linear codes in Aydins database.

cs.IT↗

arXiv2026

Entanglement-Assisted Quantum Locally Recoverable Codes: Characterizations, Bounds, and Constructions

Quantum locally recoverable codes (qLRCs) allow a single qudit erasure to be corrected by accessing only a small number of other qudits. Standard CSS and Hermitian constructions, however, impose dual-containing or self-orthogonal constraints on the underlying classical codes, thereby restricting the well-structured classical LRCs (cLRCs) that can be used to construct qLRCs. To relax these constraints, we introduce entanglement-assisted quantum locally recoverable codes (EAQLRCs) by assuming that halves of the pre-shared maximally entangled pairs are noiseless. We characterize sufficient support conditions on extended stabilizers under which entanglement-assisted stabilizer codes have locality $r$ and derive a CSS-like construction from two classical codes without imposing the ordinary dual-containing condition. We further establish an upper bound on locality and a Singleton-like bound for arbitrary CSS-like EAQLRCs, and characterize the pure codes attaining equality in the latter bound. These results yield a general framework for constructing optimal pure EAQLRCs from pairs of cLRCs. Applying this framework to $\ell$-intersection pairs of MDS codes and block parity-check matrices, we obtain two families of optimal pure CSS-like EAQLRCs with flexible parameters and nontrivial localities. To the best of our knowledge, these represent the first explicit families of EAQLRCs.

cs.IT↗

arXiv2026

Generalized Extended Codes with Applications in Entanglement-Assisted Qubit and Qutrit Codes

We prove that any generalized extended code is monomially equivalent to the Hermitian dual of a code which is closely related to a second kind of extended code of $\C^{\perp_{\rm H}}$. Every $[n+1,k+1]_{q^2}$ linear code $\D$ with $d(\D^{\perp_{\rm H}})>1$ is monomially equivalent to the generalized extended code $\C({\bf u},a)$ of an $[n,k]_{q^2}$ linear code $\C$ for a fixed $a\in\F_{q^2}^{*}$ and some ${\bf u}\in\F_{q^2}^{n}$. We then characterize the Hermitian hull and Hermitian dual distance of $\C({\bf u},a)$ in terms of the position of ${\bf u}$ relative to $\C+\C^{\perp_{\rm H}}$ and the interaction between ${\bf u}$ and the minimum weight codewords of $\C^{\perp_{\rm H}}$, respectively. We obtain explicit criteria to independently control the expected Hermitian hull dimension and Hermitian dual distance of $\C({\bf u},a)$. In particular, several conditions for simultaneously increasing the Hermitian hull dimension and the Hermitian dual distance of $\C({\bf u},a)$ are derived. Applying these results to the Hermitian construction for EAQECCs gives us $267$ new EA qubit codes of lengths $n \leq 40$ and $14$ new EA qutrit codes of lengths $n \leq 25$ compared to the best-known codes in Grassl's code tables and the imporvements recorded in very recent works in the literature. Among the new parameter sets, we confirm improvements for $236$ qubit and $8$ qutrit codes.

cs.IT↗

arXiv2026

A framework for constructing non-GRS MDS-NMDS codes from deep holes and its application

Maximum distance separable (MDS) codes and near MDS (NMDS) codes are of particular interest in coding theory due to their optimal error-correcting capabilities and wide applications in communication, cryptography, and storage systems. A family of linear codes is called a family of non-GRS MDS-NMDS codes if for each $[n,k]_q$ code in the family, it is either an $[n,k,n-k+1]_q$ MDS code that is not monomially equivalent to any GRS code or extended GRS code, or an $[n,k,n-k]_q$ NMDS code. This paper develops a unified framework for constructing new families of non-GRS MDS-NMDS codes via deep holes. We show that, starting from a family of $[n,k]_q$ non-GRS MDS-NMDS codes with covering radius $n-k$, one can systematically obtain more $[n+1,k+1]_q$ non-GRS MDS-NMDS codes. The proposed framework is further reformulated in terms of the second kind of extended codes. This reformulation recovers a main result of Wu, Ding, and Chen (IEEE Trans. Inf. Theory, 71(1): 263-272, 2025), provides a provable reduction in the computational complexity compared with the approach of Ma, Kai, and Zhu (Finite Fields Appl., 114, 102844, 2026), and reveals additional structural properties of the resulting codes. As an application, we determine the covering radius and characterize two classes of deep holes of extended subcodes of GRS codes. By applying our framework, we obtain three new families of non-GRS MDS-NMDS codes and investigate the monomial equivalence between the resulting codes and Roth-Lempel codes.

cs.IT↗

arXiv2026

Concatenated Sum-Rank Codes

Sum-rank codes have wide applications in multishot network coding, distributed storage and the construction of space-time codes. Asymptotically good sequences of linearized algebraic geometry sum-rank codes, exceeding the Gilbert-Varshamov-like bound, were constructed in a recent paper published in IEEE Trans. Inf. Theory by E. Berardini and X. Caruso. We call this bound the Tsfasman-Vladut-Zink-like bound. In this paper, we introduce the concatenation of a sum-rank code and a Hamming metric code. Then many sum-rank codes with good parameters, which are better than sum-rank BCH codes, are constructed simply and explicitly. Moreover, we obtain an asymptotically good sequence of sum-rank codes exceeding the Tsfasman-Vladut-Zink-like bound and the Gilbert-Varshamov-like bound.

cs.IT↗

arXiv2026

Linear complementary dual quasi-cyclic codes of index 2

We provide a polynomial approach to investigate linear complementary dual (LCD) quasi-cyclic codes over finite fields. We establish necessary and sufficient conditions for LCD quasi-cyclic codes of index 2 with respect to the Euclidean, Hermitian, and symplectic inner products. As a consequence of these characterizations, we derive necessary and sufficient conditions for LCD one-generator quasi-cyclic codes. Furthermore, using these characterizations, we construct some new quasi-cyclic LCD codes over small fields.

cs.IT↗

arXiv2026

Linear Complementary Pairs of Quasi-Cyclic and Quasi-Twisted Codes

In this paper, we provide a polynomial characterization of linear complementary pairs of quasi-cyclic and quasi-twisted codes of index 2. We also give several examples of linear complementary pairs of quasi-cyclic and quasi-twisted codes with optimal security parameters.

cs.IT↗

arXiv2026

Unique Decoding of Extended Subcodes of GRS Codes Using Error-Correcting Pairs

Extended Han-Zhang codes are a class of linear codes where each code is either a non-generalized Reed-Solomon (non-GRS) maximum distance separable (MDS) code or a near MDS (NMDS) code. They have important applications in communication, cryptography, and storage systems. While many algebraic properties and explicit constructions of extended Han-Zhang codes have been well studied in the literature, their decoding has been unexplored. In this paper, we focus on their decoding problems in terms of $\ell$-error-correcting pairs ($\ell$-ECPs) and deep holes. On the one hand, we determine the existence and specific forms of their $\ell$-ECPs, and further present an explicit decoding algorithm for extended Han-Zhang codes based on these $\ell$-ECPs, which can correct up to $\ell$ errors in polynomial time, with $\ell$ about half of the minimum distance. On the other hand, we determine the covering radius of extended Han-Zhang codes and characterize two classes of their deep holes, which are closely related to the maximum-likelihood decoding method. By employing these deep holes, we also construct more non-GRS MDS codes with larger lengths and dimensions, and discuss the monomial equivalence between them and the well-known Roth-Lempel codes. Some concrete examples are also given to support these results.

cs.IT↗

arXiv2025

Improved bounds and optimal constructions of pure quantum locally recoverable codes

By incorporating the concept of locality into quantum information theory, quantum locally recoverable codes (qLRCs) have been proposed, motivated by their potential applications in large-scale quantum data storage and their relevance to quantum LDPC codes. Despite the progress in optimal quantum error-correcting codes (QECCs), optimal constructions of qLRCs remain largely unexplored, partly due to the fact that the existing bounds for qLRCs are not sufficiently tight. In this paper, we focus on pure qLRCs derived from the Hermitian construction. We provide several new bounds for pure qLRCs and demonstrate that they are tighter than previously known bounds. Moreover, we show that a variety of classical QECCs, including quantum Hamming codes, quantum GRM codes, and quantum Solomon-Stiffler codes, give rise to pure qLRCs with explicit parameters. Based on these constructions, we further identify many infinite families of optimal qLRCs with respect to different bounds, achieving code lengths much larger than those of known optimal qLRCs.

cs.IT↗

arXiv2025

On optimal quantum LRCs from the Hermitian construction and $t$-designs

In a recent work, quantum locally recoverable codes (qLRCs) have been introduced for their potential application in large-scale quantum data storage and implication for quantum LDPC codes. This work focuses on the bounds and constructions of qLRCs derived from the Hermitian construction, which solves an open problem proposed by Luo $et~al.$ (IEEE Trans. Inf. Theory, 71 (3): 1794-1802, 2025). We present four bounds for qLRCs and give comparisons in terms of their asymptotic formulas. We construct several new infinite families of NMDS codes, with general and flexible dimensions, that support t-designs for $t\in \{2,3\}$, and apply them to obtain Hermitian dual-containing classical LRCs (cLRCs). As a result, we derive three explicit families of optimal qLRCs. Compared to the known qLRCs obtained by the CSS construction, our optimal qLRCs offer new and more flexible parameters. It is also worth noting that the constructed cLRCs themselves are interesting as they are optimal with respect to four distinct bounds for cLRCs.

cs.IT↗

arXiv2025

Properties and Decoding of Twisted GRS Codes and Their Extensions

Maximum distance separable (MDS) codes that are not equivalent to generalized Reed-Solomon (GRS) codes are called non-GRS MDS codes. Alongside near MDS (NMDS) codes, they are applicable in communication, cryptography, and storage systems. From theoretical perspective, it is particularly intriguing to investigate families of linear codes in which each element can be determined to be either a non-GRS MDS or an NMDS code. Two promising candidates for such families emerge from what is known as twisted GRS (TGRS) construction. These candidates are the $(+)$-TGRS codes and their extended versions, called $(+)$-extended TGRS (ETGRS) codes. Although many of their properties have been characterized, there are gaps to fill. Which among the codes are non-GRS MDS? Can we improve on their decoding by using their error-correcting pairs or deep holes? In this paper we solve these problems. The answer to the first problem leads us to two classes of non-GRS MDS Hermitian self-dual TGRS codes and a proof that there is no Galois self-dual ETGRS code. Addressing the second problem, we present an explicit decoding algorithm for ETGRS codes that outperforms existing decoding algorithms given some conditions. By considering the duals of TGRS codes which are MDS, we determine the covering radius and a class of deep holes of the recently constructed non-GRS MDS codes due to Han and Zhang.

cs.IT↗

arXiv2025

Quasi-cyclic codes of index 2

We study quasi-cyclic codes of index 2 over finite fields. We give a classification of such codes. Their duals with respect to the Euclidean, symplectic and Hermitian inner products are investigated. We describe self-orthogonal and dual-containing codes. Lower bounds for minimum distances of quasi-cyclic codes are given. A quasi-cyclic code of index 2 is generated by at most two elements. We describe conditions when such a code (or its dual) is generated by one element.

cs.IT↗

arXiv2024

A further study on the mass formula for linear codes with prescribed hull dimension

Finding a mass formula for a given class of linear codes is a fundamental problem in combinatorics and coding theory. In this paper, we consider the action of the unitary (resp. symplectic) group on the set of all Hermitian (resp. symplectic) linear complementary dual (LCD) codes, prove that all Hermitian (resp. symplectic) LCD codes are on a unique orbit under this action, and determine the formula for the size of the orbit. Based on this, we develop a general technique to obtain a closed mass formula for linear codes with prescribed Hermitian (resp. symplectic) hull dimension, and further obtain some asymptotic results.

cs.IT↗

arXiv2024

Characterization of Nearly Self-Orthogonal Quasi-Twisted Codes and Related Quantum Codes

Quasi-twisted codes are used here as the classical ingredients in the so-called Construction X for quantum error-control codes. The construction utilizes nearly self-orthogonal codes to design quantum stabilizer codes. We expand the choices of the inner product to also cover the symplectic and trace-symplectic inner products, in addition to the original Hermitian one. A refined lower bound on the minimum distance of the resulting quantum codes is established and illustrated. We report numerous record breaking quantum codes from our randomized search for inclusion in the updated online database.

quant-ph↗

arXiv2024

On Linear Codes Whose Hermitian Hulls are MD

Hermitian hulls of linear codes are interesting for theoretical and practical reasons alike. In terms of recent application, linear codes whose hulls meet certain conditions have been utilized as ingredients to construct entanglement-assisted quantum error correcting codes. This family of quantum codes is often seen as a generalization of quantum stabilizer codes. Theoretically, compared with the Euclidean setup, the Hermitian case is much harder to deal with. Hermitian hulls of MDS linear codes with low dimensions have been explored, mostly from generalized Reed-Solomon codes. Characterizing Hermitian hulls which themselves are MDS appears to be more involved and has not been extensively studied. This paper introduces some tools to study linear codes whose Hermitian hulls are MDS. Using the tools, we then propose explicit constructions of such codes. We consider Hermitian hulls of both Reed-Solomon and non Reed-Solomon types of linear MDS codes. We demonstrate that, given the same Hermitian hull dimensions, the codes from our constructions have dimensions which are larger than those in the literature.

cs.IT↗

arXiv2024

Griesmer Bound and Constructions of Linear Codes in $b$-Symbol Metric

The $b$-symbol metric is a generalization of the Hamming metric. Linear codes, in the $b$-symbol metric, have been used in the read channel whose outputs consist of $b$ consecutive symbols. The Griesmer bound outperforms the Singleton bound for $\mathbb{F}_q$-linear codes in the Hamming metric, when $q$ is fixed and the length is large enough. This scenario is also applicable in the $b$-symbol metric. Shi, Zhu, and Helleseth recently made a conjecture on cyclic codes in the $b$-symbol metric. In this paper, we present the $b$-symbol Griesmer bound for linear codes by concatenating linear codes and simplex codes. Based on cyclic codes and extended cyclic codes, we propose two families of distance-optimal linear codes with respect to the $b$-symbol Griesmer bound.

cs.IT↗