arXiv subjects

Gaojun Luo

Publications and source records attributed to Gaojun Luo.

At least 19 recordsLinked to original sources

arXiv2026

On the exact quantum chromatic number of generalized Johnson graphs

The quantum chromatic number is a fundamental parameter in the study of nonlocal games, capturing the extent to which entanglement can improve performance in distributed tasks. In this paper, we investigate the quantum chromatic number of generalized Johnson graphs. By constructing modulus-one orthogonal representations, we obtain general upper bounds on their quantum chromatic numbers. We further analyze the smallest eigenvalue of these graphs. Combining the resulting Hoffman-type lower bounds with the upper bounds obtained from orthogonal representations, we determine the exact quantum chromatic numbers of two infinite families of generalized Johnson graphs. Finally, applying a forbidden-distance theorem for binary codes, we show that the classical chromatic numbers of these families grow exponentially with $n$, whereas their quantum chromatic numbers grow linearly. These families exhibit an exponential separation between the classical and quantum chromatic numbers.

math.CO↗

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

Robust Repair of Reed-Solomon Codes

We study the problem of robust repair of a single erasure in Reed--Solomon codes under low communication bandwidth. Focusing on the Guruswami--Wootters trace repair framework, we investigate whether a failed node can be correctly repaired in the presence of erroneous responses from helper nodes. Equivalently, we view the collection of downloaded traces as a code, which we call the repair-trace code. By characterizing the zero coefficients of the associated polynomial in terms of cyclotomic cosets, we derive upper bounds on the dimension $k$ that allow correction of a given number of erroneous traces $e$, as well as lower bounds on the minimum distance as a function of $k$. For the case $q=2$, we exploit explicit formulas for cyclotomic coset representatives to obtain the exact optimal dimension bound for single-error correction. We also propose two efficient robust repair schemes. Our first scheme achieves the error-correction capability guaranteed by the BCH bound. To approach a stronger bound based on character sums, we develop a second scheme that tolerates more errors at the cost of an additional factor $n$ in computational complexity.

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

Generalized Repetition Codes and Their Application to HARQ

The inherent uncertainty of communication channels implies that any coding scheme has a non-zero probability of failing to correct errors, making retransmission mechanisms essential. To ensure message reliability and integrity, a dual-layer redundancy framework is typically employed: error correction codes mitigate noise-induced impairments at the physical layer, while cyclic redundancy checks verify message integrity after decoding. Retransmission is initiated if verification fails. This operational model can be categorized into two types of repeated communication models: Type-I systems repeatedly transmit identical codewords, whereas Type-II systems transmit distinct coded representations of the same message. The core challenge lies in maximizing the probability of correct message decoding within a limited number of transmission rounds through verification-based feedback mechanisms. In this paper, we consider a scenario where the same error-correcting code is used for repeated transmissions, and we specifically propose two classes of generalized repetition codes (GRCs), corresponding to the two repeated communication models. In contrast to classical theory, we regard GRCs as error-correcting codes under multiple metrics--that is, GRCs possess multiple minimum distances. This design enables GRCs to perform multi-round error correction under different metrics, achieving stronger error-correction capabilities than classical error-correcting codes. However, the special structure of GRCs makes their construction more challenging, as it requires simultaneously optimizing multiple minimum distances. To address this, we separately investigate the bounds and constructions for Type-I and Type-II GRCs, and obtain numerous optimal Type-I and Type-II GRCs.

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

Lower Bounds for Error Coefficients of Griesmer Optimal Linear Codes via Iteration

The error coefficient of a linear code is defined as the number of minimum-weight codewords. In an additive white Gaussian noise channel, optimal linear codes with the smallest error coefficients achieve the best possible asymptotic frame error rate (AFER) among all optimal linear codes under maximum likelihood decoding. Such codes are referred to as AFER-optimal linear codes. The Griesmer bound is essential for determining the optimality of linear codes. However, establishing tight lower bounds on the error coefficients of Griesmer optimal linear codes is challenging, and the linear programming bound often performs inadequately. In this paper, we propose several iterative lower bounds for the error coefficients of Griesmer optimal linear codes. Specifically, for binary linear codes, our bounds are tight in most cases when the dimension does not exceed $5$. To evaluate the performance of our bounds when they are not tight, we also determine the parameters of the remaining 5-dimensional AFER-optimal linear codes. Our final comparison demonstrates that even when our bounds are not tight, they remain very close to the actual values, with a gap of less than or equal to $2$.

cs.IT↗

arXiv2025

Entanglement-Assisted Concatenated Quantum Codes: Parameters and Asymptotic Performance

Entanglement-assisted concatenated quantum codes (EACQCs) are constructed by concatenating two entanglement-assisted quantum error-correcting codes (EAQECCs). By selecting the inner and outer component codes carefully, it is able to construct state-of-the-art EACQCs with parameters better than previous quantum codes. In this work, we use almost maximum-distance-separable (MDS) codes and $\hbar$-MDS codes as the outer codes to construct EACQCs. Because the range of code length of almost MDS and $\hbar$-MDS codes is much more free than that of the commonly used MDS codes. We derive several families of new EACQCs with parameters better than the previously best known EAQECCs and standard quantum error-correcting codes (QECCs) of the same length and net transmissions. Moreover, we demonstrate that EACQCs are with maximal entanglement if both the inner and outer component codes are with maximal entanglement. As a result, we construct three new maximal-entanglement EACQCs which have optimal parameters. In addition, we present several new maximal-entanglement EACQCs whose minimum distance is only one less than the minimum distance of the optimal codes. In particular, we propose two new families of asymptotically good maximal-entanglement EACQCs with explicit constructions by using entanglement-assisted quantum algebraic geometry codes as the outer codes. At last, we prove that EACQCs can attain the quantum Gilbert-Varshamov bound for EAQECCs asymptotically.

quant-ph↗

arXiv2024

Combinatorial Constructions of Optimal Quaternary Additive Codes

This paper aims to construct optimal quaternary additive codes with non-integer dimensions. Firstly, we propose combinatorial constructions of quaternary additive constant-weight codes, alongside additive generalized anticode construction. Subsequently, we propose generalized Construction X, which facilitates the construction of non-integer dimensional optimal additive codes from linear codes. Then, we construct ten classes of optimal quaternary non-integer dimensional additive codes through these two methods. As an application, we also determine the optimal additive $[n,3.5,n-t]_4$ codes for all $t$ with variable $n$, except for $t=6,7,12$.

cs.IT↗

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↗

arXiv2023

Bounds and Constructions of Quantum Locally Recoverable Codes from Quantum CSS Codes

Classical locally recoverable codes (LRCs) have become indispensable in distributed storage systems. They provide efficient recovery in terms of localized errors. Quantum LRCs have very recently been introduced for their potential application in quantum data storage. In this paper, we use classical LRCs to investigate quantum LRCs. We prove that the parameters of quantum LRCs are bounded by their classical counterparts. We deduce the bounds on the parameters of quantum LRCs from the bounds on the parameters of the classical ones. We establish a characterization of optimal pure quantum LRCs based on classical codes with specific properties. Using well-crafted classical LRCs as ingredients in the construction of quantum CSS codes, we offer the first construction of several families of optimal pure quantum LRCs.

cs.IT↗

arXiv2023

Improved Spectral Bound for Quasi-Cyclic Codes

Spectral bounds form a powerful tool to estimate the minimum distances of quasi-cyclic codes. They generalize the defining set bounds of cyclic codes to those of quasi-cyclic codes. Based on the eigenvalues of quasi-cyclic codes and the corresponding eigenspaces, we provide an improved spectral bound for quasi-cyclic codes. Numerical results verify that the improved bound outperforms the Jensen bound in almost all cases. Based on the improved bound, we propose a general construction of quasi-cyclic codes with excellent designed minimum distances. For the quasi-cyclic codes produced by this general construction, the improved spectral bound is always sharper than the Jensen bound.

cs.IT↗

arXiv2023

On non-expandable cross-bifix-free codes

A cross-bifix-free code of length $n$ over $\mathbb{Z}_q$ is defined as a non-empty subset of $\mathbb{Z}_q^n$ satisfying that the prefix set of each codeword is disjoint from the suffix set of every codeword. Cross-bifix-free codes have found important applications in digital communication systems. One of the main research problems on cross-bifix-free codes is to construct cross-bifix-free codes as large as possible in size. Recently, Wang and Wang introduced a family of cross-bifix-free codes $S_{I,J}^{(k)}(n)$, which is a generalization of the classical cross-bifix-free codes studied early by Lvenshtein, Gilbert and Chee {\it et al.}. It is known that $S_{I,J}^{(k)}(n)$ is nearly optimal in size and $S_{I,J}^{(k)}(n)$ is non-expandable if $k=n-1$ or $1\leq k<n/2$. In this paper, we first show that $S_{I,J}^{(k)}(n)$ is non-expandable if and only if $k=n-1$ or $1\leq k<n/2$, thereby improving the results in [Chee {\it et al.}, IEEE-TIT, 2013] and [Wang and Wang, IEEE-TIT, 2022]. We then construct a new family of cross-bifix-free codes $U^{(t)}_{I,J}(n)$ to expand $S_{I,J}^{(k)}(n)$ such that the resulting larger code $S_{I,J}^{(k)}(n)\bigcup U^{(t)}_{I,J}(n)$ is a non-expandable cross-bifix-free code whenever $S_{I,J}^{(k)}(n)$ is expandable. Finally, we present an explicit formula for the size of $S_{I,J}^{(k)}(n)\bigcup U^{(t)}_{I,J}(n)$.

cs.IT↗

arXiv2023

A Construction of Asymptotically Optimal Cascaded CDC Schemes via Combinatorial Designs

A coded distributed computing (CDC) system aims to reduce the communication load in the MapReduce framework. Such a system has $K$ nodes, $N$ input files, and $Q$ Reduce functions. Each input file is mapped by $r$ nodes and each Reduce function is computed by $s$ nodes. The objective is to achieve the maximum multicast gain. There are known CDC schemes that achieve optimal communication load. In some prominent known schemes, however, $N$ and $Q$ grow too fast in terms of $K$, greatly reducing their gains in practical scenarios. To mitigate the situation, some asymptotically optimal cascaded CDC schemes with $r=s$ have been proposed by using symmetric designs. In this paper, we put forward new asymptotically optimal cascaded CDC schemes with $r=s$ by using $1$-designs. Compared with earlier schemes from symmetric designs, ours have much smaller computation loads while keeping the other relevant parameters the same. We also obtain new asymptotically optimal cascaded CDC schemes with more flexible parameters compared with previously best-performing schemes.

cs.IT↗

arXiv2023

Sharper Asymptotically Optimal CDC Schemes via Combinatorial Designs

Coded distributed computing (CDC) was introduced to greatly reduce the communication load for MapReduce computing systems. Such a system has $K$ nodes, $N$ input files, and $Q$ Reduce functions. Each input file is mapped by $r$ nodes and each Reduce function is computed by $s$ nodes. The architecture must allow for coding techniques that achieve the maximum multicast gain. Some CDC schemes that achieve optimal communication load have been proposed before. The parameters $N$ and $Q$ in those schemes, however, grow too fast with respect to $K$ to be of great practical value. To improve the situation, researchers have come up with some asymptotically optimal cascaded CDC schemes with $s+r=K$ from symmetric designs. In this paper, we propose new asymptotically optimal cascaded CDC schemes. Akin to known schemes, ours have $r+s=K$ and make use of symmetric designs as construction tools. Unlike previous schemes, ours have much smaller communication loads, given the same set of parameters $K$, $r$, $N$, and $Q$. We also expand the construction tools to include almost difference sets. Using them, we have managed to construct a new asymptotically optimal cascaded CDC scheme.

cs.IT↗

arXiv2023

Repair of Reed-Solomon Codes in the Presence of Erroneous Nodes

We consider the repair scheme of Guruswami-Wootters for the Reed-Solomon code and ask: can we correctly repair a failed node in the presence of erroneous nodes? Equivalently, we consider the collection of downloaded traces as a code and investigate its code-distance properties. We propose three lower bounds on its minimum distance and study methods to efficiently correct errors close to these bounds.

cs.IT↗

arXiv2022

Sharp Uncertainty Principle for Transitive $G$-Sets over Arbitrary Fields and Finite Groups

For any finite group $G$, any transitive $G$-set $X$ and any field ${\Bbb F}$, we consider the vector space ${\Bbb F}^X$ of all functions from $X$ to ${\Bbb F}$, which is a $G$-space isomorphic to the permutation ${\Bbb F} G$-module ${\Bbb F} X$. When the group algebra ${\Bbb F} G$ is semisimple and split, we find a specific basis $\widehat X$ of ${\Bbb F}^X$ and, for $f\in{\Bbb F}^X$, construct the Fourier transform $\widehat f\in{\Bbb F}^{\widehat X}$. We define the rank support $\mbox{rk-supp}(\widehat f)$ and prove that $\mbox{rk-supp}(\widehat f)=\dim {\Bbb F} G f$, where ${\Bbb F} G f$ is the submodule of ${\Bbb F} X$ generated by the element $f=\sum_{x\in X}f(x)x$. Next, we extend and strengthen the sharpened uncertainty principle for finite abelian groups, established by Feng, Hollmann, and Xiang in 2019, to a broader framework and a sharp version. For $0\ne f\in{\Bbb F}^X$, we construct a block $X_{{\rm supp}(f)}$ of $X$ and a subset ${\mathscr S}'^{-\!1}$ of $G$ determined by the support ${\rm supp}(f)$ of $f$, and show that $\dim{\Bbb F} Gf-\dim{\Bbb F}{\mathscr S}'^{-\!1}\!f\ge 1$ and $$ |{\rm supp}(f)|\cdot \dim{\Bbb F} Gf \ge |X|+ (\!\dim{\Bbb F} Gf-\dim{\Bbb F}{\mathscr S}'^{-1}f) \cdot|{\rm supp}(f)| -|X_{{\rm supp}(f)}|, $$ where ${\Bbb F}{\mathscr S}'^{-1}f$ denotes the subspace of ${\Bbb F}X$ spanned by the subset ${\mathscr S}'^{-1}f=\{\alpha f\,|\,\alpha\in{\mathscr S}'^{-1}\}\subseteq{\Bbb F} X$. We provide necessary and sufficient conditions for the above inequality to achieve equality. As corollaries, we derive many sharpened or classical versions of the finite-dimensional uncertainty principle, address an open question posed by Feng, Hollmann, and Xiang. When $|G|$ is a prime and $X=G$, we give a lower bound on $\dim {\Bbb F}Gf$ that recovers Tao's 2005 strong uncertainty principle, along with a precise characterization of the equality case.

math.GR↗