arXiv subjects

Weijun Fang

Publications and source records attributed to Weijun Fang.

At least 19 recordsLinked to original sources

arXiv2026

Duality and Reverse Self-Dual Constructions for Hyperderivative Reed-Solomon Codes

Hyperderivative Reed-Solomon (HRS) codes form a class of maximum-distance-separable codes under the Niederreiter-Rosenbloom-Tsfasman metric and may be viewed as a derivative-evaluation extension of classical Reed-Solomon codes. For generalized Reed-Solomon codes, the Euclidean dual is again a generalized Reed-Solomon code. In this paper, we investigate the corresponding duality problem for HRS codes. Using a residue-theoretic argument, we derive an explicit component-wise representation for the Euclidean dual of an HRS code. The formula shows that, in general, the Euclidean dual is not an HRS code. Instead, it is blockwise upper-triangularly equivalent to a reverse-order HRS evaluation code, where the reversal occurs in the hyperderivative orders within each evaluation block. In particular, for full-domain HRS codes with low multiplicity, the triangular transformations reduce to diagonal scalings, and the Euclidean dual is obtained as the row reversal of an HRS code. Based on this reverse-order dual structure, we further study reverse self-dual HRS codes. We establish explicit criteria for reverse self-duality and construct several families from additive and multiplicative coset structures.

cs.IT↗

arXiv2026

Tight Bandwidth Lower Bounds and Optimal Constructions of Locally Repairable Convertible Codes in the Global Split Regime

Erasure coding is a key technique for providing fault tolerance in modern distributed storage systems. In practice, as storage systems evolve, the parameters of the deployed erasure code may need to be adjusted to accommodate changes in storage scale, reliability requirements, and disk failure rates. Such adaptation is achieved through code conversion, which transforms data encoded by an initial code into data encoded by a final code. Convertible codes are designed to carry out this transformation efficiently while preserving desirable code properties. In this work, we study code conversion between systematic optimal-distance locally repairable codes (LRCs) in the global split regime, using read bandwidth as the conversion-efficiency metric. Specifically, we focus on the parameter range $g^I,g^F \leq r$, where the numbers of initial and final global parity nodes are at most the local information dimension $r$. Over this entire parameter range, we derive lower bounds on the read bandwidth of stable optimal-distance locally repairable convertible codes (LRCCs) via an information-theoretic approach, without imposing any linearity assumption on the initial codes, the final codes, or the conversion procedure. We then develop constructions based on MDS array codes with prescribed repair or alignment properties. Depending on the relative sizes of $g^I$ and $g^F$, we handle the construction separately in the three cases $g^F=g^I$, $g^F>g^I$, and $g^F<g^I$, and show that each attains the corresponding lower bound. This yields a complete characterization of the optimal read bandwidth for stable optimal-distance LRCCs over the entire parameter range $g^I,g^F\le r$.

cs.IT↗

arXiv2026

Private Information Retrieval from Joint Systematic MDS-Coded with Non-Colluding Servers: Bounds and Constructions

Consider a distributed storage system consisting of $N$ non-colluding servers that collectively store a database of $M$ files encoded using an $[N,K]$ maximum distance separable(MDS) code. A user wishes to retrieve one file privately by accessing the servers without revealing the identity of the requested file. A scheme designed for this purpose is called a joint MDS-coded private information retrieval(PIR) scheme, which was first introduced by Sun and Tian in 2019 to break the capacity $\frac{1-K/N}{1-(K/N)^M}$ of the separate MDS-coded PIR schemes established by Banawan and Ulukus. However, the capacity of joint MDS-coded PIR remains largely unexplored. In this paper, we study the capacity of joint MDS-coded PIR with systematic MDS array storage codes under prescribed storage patterns. Specifically, we first derive upper bounds on the capacity of joint MDS-coded PIR for $K=Mt$ and $K=Mt+1$, respectively. We then construct three joint MDS-coded PIR schemes for the cases $N\le K+t, K=Mt$, $N>K+t, K=Mt$ and $N\le K+t, K=Mt+1$. The proposed schemes require small file sizes and achieve higher retrieval rates: the first and third schemes exceed the capacity of separate MDS-coded PIR schemes, while the second scheme does so when the storage rate $\frac{K}{N}>r_M$ for some $0<r_M<\frac{M}{M+1}$. In particular, for $K=Mt$ and $N\leq K+t$, the proposed scheme achieves the derived upper bound, thereby establishing that the optimal joint MDS-coded PIR capacity under the considered storage pattern is $1-(1-\frac{1}{M})\frac{K}{N}$. Compared with capacity-achieving separate MDS-coded PIR schemes at the same storage-code rate, the proposed schemes may achieve a substantial relative retrieval-rate improvement: the maximum improvement can exceed $15\%$ when $M\geq 4$, exceed $20\%$ when $M\geq 9$, and asymptotically approach $1-2/e\approx 26.42\%$ as M increases.

cs.IT↗

arXiv2026

Reed-Solomon Codes with Optimal Repair Bandwidth: A Basis-Transformation Approach

Maximum distance separable (MDS) codes are widely used in distributed storage, but naively repairing a single failure in an $(n,k)$ MDS code requires downloading the full contents of $k$ surviving nodes. Minimum storage regenerating (MSR) codes, introduced by Dimakis et al., minimize repair bandwidth while preserving the MDS property by contacting $d>k$ helper nodes and downloading only a fraction of each helper. For scalar MDS codes, Guruswami and Wootters established a linear repair framework, and Tamo, Ye, and Barg subsequently gave the first explicit Reed-Solomon (RS) codes achieving the MSR point. Their construction yields RS-MSR codes with subpacketization $\ell=s\prod_{i=1}^n p_i$, where $s=d+1-k$ and the distinct primes $p_i$ satisfy $p_i\equiv 1\pmod{s}$. In this paper, we show that this congruence condition is not intrinsic to the RS repair problem. We develop a basis-transformation approach to the construction of repair-enabling subspaces. The approach consists of three deterministic operations -- Euclidean Square Partition, Transposition, and Column Aggregation -- which construct the required repair-enabling subspaces directly from the standard monomial basis of the repair field. Consequently, we obtain RS-MSR codes with subpacketization $\ell=s\prod_{i=1}^n p_i$ for arbitrary distinct primes $p_i>s$. For fixed $s$, this improves the subpacketization of the Tamo--Ye--Barg construction by a factor asymptotic to $\varphi(s)^{n+\mathrm{o}(n)}$, where $\varphi(\cdot)$ denotes Euler's totient function.

cs.IT↗

arXiv2026

New $X$-Secure $T$-Private Information Retrieval Schemes via Rational Curves and Hermitian Curves

$X$-secure and $T$-private information retrieval (XSTPIR) is a variant of private information retrieval where data security is guaranteed against collusion among up to $X$ servers and the user's retrieval privacy is guaranteed against collusion among up to $T$ servers. Recently, researchers have constructed XSTPIR schemes through the theory of algebraic geometry codes and algebraic curves, with the aim of obtaining XSTPIR schemes that have higher maximum PIR rates for fixed field size and $X,T$ (the number of servers $N$ is not restricted). The mainstream approach is to employ curves of higher genus that have more rational points, evolving from rational curves to elliptic curves to hyperelliptic curves and, most recently, to Hermitian curves. In this paper, we propose a different perspective: with the shared goal of constructing XSTPIR schemes with higher maximum PIR rates, we move beyond the mainstream approach of seeking curves with higher genus and more rational points. Instead, we aim to achieve this goal by enhancing the utilization efficiency of rational points on curves that have already been considered in previous work. By introducing a family of bases for the polynomial space $\text{span}_{\mathbb{F}_q}\{1,x,\dots,x^{k-1}\}$ as an alternative to the Lagrange interpolation basis, we develop two new families of XSTPIR schemes based on rational curves and Hermitian curves, respectively. Parameter comparisons demonstrate that our schemes achieve superior performance. Specifically, our Hermitian-curve-based XSTPIR scheme provides the largest known maximum PIR rates when the field size $q^2\geq 14^2$ and $X+T\geq 4q$. Moreover, for any field size $q^2\geq 28^2$ and $X+T\geq 4$, our two XSTPIR schemes collectively provide the largest known maximum PIR rates.

cs.IT↗

arXiv2026

Some New Results on Sequence Reconstruction Problem for Deletion Channels

Levenshtein first introduced the sequence reconstruction problem in $2001$. In the realm of combinatorics, the sequence reconstruction problem is equivalent to determining the value of $N(n,d,t)$, which represents the maximum size of the intersection of two metric balls of radius $t$, given that the distance between their centers is at least $d$ and the sequence length is $n$. In this paper, We present a lower bound on $N(n,3,t)$ for $n\geq \max\{13,t+8\}$ and $t \geq 4$. For $t=4$, we prove that this lower bound is tight. This settles an open question posed by Pham, Goyal, and Kiah, confirming that $N(n,3,4)=20n-166$ for all $n \geq 13$.

cs.IT↗

arXiv2025

Making Wide Stripes Practical: Cascaded Parity LRCs for Efficient Repair and High Reliability

Erasure coding with wide stripes is increasingly adopted to reduce storage overhead in large-scale storage systems. However, existing Locally Repairable Codes (LRCs) exhibit structural limitations in this setting: inflated local groups increase single-node repair cost, multi-node failures frequently trigger expensive global repair, and reliability degrades sharply. We identify a key root cause: local and global parity blocks are designed independently, preventing them from cooperating during repair. We present Cascaded Parity LRCs (CP-LRCs), a new family of wide stripe LRCs that embed structured dependency between parity blocks by decomposing a global parity block across all local parity blocks. This creates a cascaded parity group that preserves MDS-level fault tolerance while enabling low-bandwidth single-node and multi-node repairs. We provide a general coefficient-generation framework, develop repair algorithms exploiting cascading, and instantiate the design with CP-Azure and CP-Uniform. Evaluations on Alibaba Cloud show reductions in repair time of up to 41% for single-node failures and 26% for two-node failures.

cs.DC↗

arXiv2025

Three Classes of Twisted Gabidulin Codes with Different Twists

Twisted Gabidulin codes are an extension of Gabidulin codes and have recently attracted great attention. In this paper, we study three classes of twisted Gabidulin codes with different twists. Moreover, we establish necessary and sufficient conditions for them to be maximum rank distance (MRD) codes, determine the conditions under which they are not MRD codes, and construct several classes of MRD codes via twisted Gabidulin codes. In addition, considering these codes in the Hamming metric, we provide necessary and sufficient conditions for them to be maximum distance separable (MDS), almost MDS, or near MDS. Finally, we investigate the covering radii and deep holes of twisted Gabidulin codes.

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arXiv2025

New Constructions of Optimal $(r,\delta)$-LRCs via Algebraic Function Fields

Constructing optimal $(r,\delta)$-LRCs that attain the Singleton-type bound is an active and important research direction, particularly due to their practical applications in distributed storage systems. In this paper, we focus on the construction of optimal $(r,\delta)$-LRCs with flexible minimum distances, especially for the case $\delta \geq 3$. We first extend a general framework -- originally proposed by Li \textit{et al.} (IEEE Trans. Inf. Theory, vol. 65, no. 1, 2019) and Ma and Xing (J. Comb. Theory Ser. A., vol. 193, 2023) -- for constructing optimal $r$-LRCs via automorphism groups of elliptic function fields to the case of $(r,\delta)$-LRCs. This newly extended general framework relies on certain conditions concerning the group law of elliptic curves. By carefully selecting elliptic function fields suitable for this framework, we arrive at several families of explicit $q$-ary optimal $(r,3)$-LRCs and $(2,\delta)$-LRCs with lengths slightly less than $q + 2\sqrt{q}$. Next, by employing automorphism groups of hyperelliptic function fields of genus $2$, we develop a framework for constructing optimal $(r,3)$-LRCs and obtain a family of explicit $q$-ary optimal $(4,3)$-LRCs with code lengths slightly below $q+4\sqrt{q}$. We then consider the construction of optimal $(r,\delta)$-LRCs via hyperelliptic function fields of arbitrary genus $g \geq 2$, yielding a class of explicit $q$-ary optimal $(g+1-g',g+1+g')$-LRCs for $0 \leq g' \leq g-1$ with lengths up to $q + 2g\sqrt{q}$. Finally, applying certain superelliptic curves derived from modified Norm-Trace curves, we construct two families of explicit optimal $(r,\delta)$-LRCs with even longer code lengths and more flexible parameters. Notably, many of the newly constructed optimal $(r,\delta)$-LRCs attain the largest known lengths among existing constructions with flexible minimum distances.

cs.IT↗

arXiv2025

Bounds and Optimal Constructions of Generalized Merge-Convertible Codes for Code Conversion into LRCs

Error-correcting codes are essential for ensuring fault tolerance in modern distributed data storage systems. However, in practice, factors such as the failure rates of storage devices can vary significantly over time, resulting in changes to the optimal code parameters. To reduce storage cost while maintaining efficiency, Maturana and Rashmi introduced a theoretical framework known as code conversion, which enables dynamic adjustment of code parameters according to device performance. In this paper, we focus exclusively on the bounds and constructions of generalized merge-convertible codes. First, we establish a new lower bound on the access cost when the final code is an $(r,δ)$-LRC. This bound unifies and generalizes all previously known bounds for merge conversion, where the initial and final codes are either LRCs or MDS codes. We then construct a family of access-optimal MDS convertible codes by leveraging subgroups of the automorphism group of a rational function field. It is worth noting that our construction is also per-symbol read access-optimal. Next, we further extend our MDS-based construction to design access-optimal convertible codes for the conversion between $(r,δ)$-LRCs with parameters that have not been previously reported. Finally, using the parity-check matrix approach, we present a construction of access-optimal convertible codes that enable merge conversion from MDS codes to an $(r,δ)$-LRC. To the best of our knowledge, this is the first explicit optimal construction of code conversion between MDS codes and LRCs. All of our constructions are performed over finite fields whose sizes grow linearly with the code length.

cs.IT↗

arXiv2025

New Constructions of Binary Cyclic Codes with Both Relatively Large Minimum Distance and Dual Distance

Binary cyclic codes are worth studying due to their applications and theoretical importance. It is an important problem to construct an infinite family of cyclic codes with large minimum distance $d$ and dual distance $d^{\perp}$. In recent years, much research has been devoted to improving the lower bound on $d$, some of which have exceeded the square-root bound. The constructions presented recently seem to indicate that when the minimum distance increases, the minimum distance of its dual code decreases. In this paper, we focus on the new constructions of binary cyclic codes with length $n=2^m-1$, dimension near $n/2$ and both relatively large minimum distance and dual distance. When $m$ is even, we construct a family of binary cyclic codes with parameters $[2^m-1,2^{m-1}\pm1,d]$, where $d\ge 2^{m/2}-1$ and $d^\perp\ge2^{m/2}$. Both the minimum distance and the dual distance are significantly better than the previous results. When $m$ is the product of two distinct primes, we construct some cyclic codes with dimensions $k=(n+1)/2$ and $d>\frac{n}{\log_2n},$ where the lower bound on the minimum distance is much larger than the square-root bound. When $m$ is odd, we present two families of binary $[2^m-1,2^{m-1},d]$ cyclic codes with $d\ge2^{(m+1)/2}-1$, $d^\perp\ge2^{(m+1)/2}$ and $d\ge2^{(m+3)/2}-15$, $d^\perp\ge2^{(m-1)/2}$ respectively, which leads that $d\cdot d^\perp$ can reach $2n$ asymptotically. To the best of our knowledge, for the binary cyclic codes with length $n=2^m-1$ and dimension $k=(n\pm1)/2$, except for the punctured binary Reed-Muller codes, there is no other construction of binary cyclic codes that reaches this bound.

cs.IT↗

arXiv2025

Construction of MDS Euclidean Self-Dual Codes via Multiple Subsets

MDS self-dual codes have good algebraic structure, and their parameters are completely determined by the code length. In recent years, the construction of MDS Euclidean self-dual codes with new lengths has become an important issue in coding theory. In this paper, we are committed to constructing new MDS Euclidean self-dual codes via generalized Reed-Solomon (GRS) codes and their extended (EGRS) codes. The main effort of our constructions is to find suitable subsets of finite fields as the evaluation sets, ensuring that the corresponding (extended) GRS codes are Euclidean self-dual. Firstly, we present a method for selecting evaluation sets from multiple intersecting subsets and provide a theorem to guarantee that the chosen evaluation sets meet the desired criteria. Secondly, based on this theorem, we construct six new classes of MDS Euclidean self-dual codes using the norm function, as well as the union of three multiplicity subgroups and their cosets respectively. Finally, in our constructions, the proportion of possible MDS Euclidean self-dual codes exceeds 85\%, which is much higher than previously reported results.

cs.IT↗

arXiv2025

New constructions of MDS symbol-pair codes via simple-root cyclic codes

In modern storage technologies, symbol-pair codes have emerged as a crucial framework for addressing errors in channels where symbols are read in overlapping pairs to guard against pair errors. A symbol-pair code that meets the Singleton-type bound is called a maximum distance separable (MDS) symbol-pair code. MDS symbol-pair codes are optimal in the sense that they have the highest pair error-correcting capability. In this paper, we focus on new constructions of MDS symbol-pair codes using simple-root cyclic codes. Specifically, three new infinite families of $(n, d_P)_q$-MDS symbol-pair codes are obtained: (1) $(n=4q+4,d_P=7)_q$ for $q\equiv 1\pmod 4$; (2) $(n=4q-4,d_P=8)_q$ for $q\equiv 3\pmod 4$; (3) $(n=2q+2,d_P=9)_q$ for $q$ being an odd prime power. The first two constructions are based on analyzing the solutions of certain equations over finite fields. The third construction arises from the decomposition of cyclic codes, where we utilize the orthogonal relationships between component codes and their duals to rigorously exclude the presence of specific codewords. It is worth noting that for the pair distance $d_P=7$ or $8$, our $q$-ary MDS symbol-pair codes achieve the longest known code length when $q$ is not a prime. Furthermore, for $d_P=9$, our codes attain the longest code length regardless of whether $q$ is prime or not.

cs.IT↗

arXiv2024

Efficiently Achieving Secure Model Training and Secure Aggregation to Ensure Bidirectional Privacy-Preservation in Federated Learning

Bidirectional privacy-preservation federated learning is crucial as both local gradients and the global model may leak privacy. However, only a few works attempt to achieve it, and they often face challenges such as excessive communication and computational overheads, or significant degradation of model accuracy, which hinders their practical applications. In this paper, we design an efficient and high-accuracy bidirectional privacy-preserving scheme for federated learning to complete secure model training and secure aggregation. To efficiently achieve bidirectional privacy, we design an efficient and accuracy-lossless model perturbation method on the server side (called $\mathbf{MP\_Server}$) that can be combined with local differential privacy (LDP) to prevent clients from accessing the model, while ensuring that the local gradients obtained on the server side satisfy LDP. Furthermore, to ensure model accuracy, we customize a distributed differential privacy mechanism on the client side (called $\mathbf{DDP\_Client}$). When combined with $\mathbf{MP\_Server}$, it ensures LDP of the local gradients, while ensuring that the aggregated result matches the accuracy of central differential privacy (CDP). Extensive experiments demonstrate that our scheme significantly outperforms state-of-the-art bidirectional privacy-preservation baselines (SOTAs) in terms of computational cost, model accuracy, and defense ability against privacy attacks. Particularly, given target accuracy, the training time of SOTAs is approximately $200$ times, or even over $1000$ times, longer than that of our scheme. When the privacy budget is set relatively small, our scheme incurs less than $6\%$ accuracy loss compared to the privacy-ignoring method, while SOTAs suffer up to $20\%$ accuracy loss. Experimental results also show that the defense capability of our scheme outperforms than SOTAs.

cs.LG↗

arXiv2024

Optimal $(2,δ)$ Locally Repairable Codes via Punctured Simplex Codes

Locally repairable codes (LRCs) have attracted a lot of attention due to their applications in distributed storage systems. In this paper, we provide new constructions of optimal $(2, δ)$-LRCs over $\mathbb{F}_q$ with flexible parameters. Firstly, employing techniques from finite geometry, we introduce a simple yet useful condition to ensure that a punctured simplex code becomes a $(2, δ)$-LRC. It is worth noting that this condition only imposes a requirement on the size of the puncturing set. Secondly, utilizing character sums over finite fields and Krawtchouk polynomials, we determine the parameters of more punctured simplex codes with puncturing sets of new structures. Several infinite families of LRCs with new parameters are derived. All of our new LRCs are optimal with respect to the generalized Cadambe-Mazumdar bound and some of them are also Griesmer codes or distance-optimal codes.

cs.IT↗

arXiv2024

Deep Holes of Twisted Reed-Solomon Codes

The deep holes of a linear code are the vectors that achieve the maximum error distance (covering radius) to the code. {Determining the covering radius and deep holes of linear codes is a fundamental problem in coding theory. In this paper, we investigate the problem of deep holes of twisted Reed-Solomon codes.} The covering radius and a standard class of deep holes of twisted Reed-Solomon codes ${\rm TRS}_k(\mathcal{A}, \theta)$ are obtained for a general evaluation set $\mathcal{A} \subseteq \mathbb{F}_q$. Furthermore, we consider the problem of determining all deep holes of the full-length twisted Reed-Solomon codes ${\rm TRS}_k(\mathbb{F}_q, \theta)$. For even $q$, by utilizing the polynomial method and Gauss sums over finite fields, we prove that the standard deep holes are all the deep holes of ${\rm TRS}_k(\mathbb{F}_q, \theta)$ with $\frac{3q-4}{4} \leq k\leq q-4$. For odd $q$, we adopt a different method and employ the results on some equations over finite fields to show that there are also no other deep holes of ${\rm TRS}_k(\mathbb{F}_q, \theta)$ with $\frac{3q+3\sqrt{q}-7}{4} \leq k\leq q-4$. In addition, for the boundary cases of $k=q-3, q-2$ and $q-1$, we completely determine their deep holes using results on certain character sums.

cs.IT↗

arXiv2023

New Lower Bounds for the Minimum Distance of Cyclic Codes and Applications to Locally Repairable Codes

Cyclic codes are an important class of linear codes. Bounding the minimum distance of cyclic codes is a long-standing research topic in coding theory, and several well-known and basic results have been developed on this topic. Recently, locally repairable codes (LRCs) have attracted much attention due to their repair efficiency in large-scale distributed storage systems. In this paper, by employing the singleton procedure technique, we first provide a sufficient condition for bounding the minimum distance of cyclic codes with typical defining sets. Secondly, by considering a specific case, we establish a connection between bounds for the minimum distance of cyclic codes and solutions to a system of inequalities. This connection leads to the derivation of new bounds, including some with general patterns. In particular, we provide three new bounds with general patterns, one of which serves as a generalization of the Betti-Sala bound. Finally, we present a generalized lower bound for a special case and construct several families of $(2, δ)$-LRCs with unbounded length and minimum distance $2δ$. It turns out that these LRCs are distance-optimal, and their parameters are new. To the best of our knowledge, this work represents the first construction of distance-optimal $(r, δ)$-LRCs with unbounded length and minimum distance exceeding $r+δ-1$.

cs.IT↗

arXiv2023

Singleton-Optimal LRCs and Perfect LRCs via Cyclic and Constacyclic Codes

Locally repairable codes (LRCs) have emerged as an important coding scheme in distributed storage systems (DSSs) with relatively low repair cost by accessing fewer non-failure nodes. Theoretical bounds and optimal constructions of LRCs have been widely investigated. Optimal LRCs via cyclic and constacyclic codes provide significant benefit of elegant algebraic structure and efficient encoding procedure. In this paper, we continue to consider the constructions of optimal LRCs via cyclic and constacyclic codes with long code length. Specifically, we first obtain two classes of $q$-ary cyclic Singleton-optimal $(n, k, d=6;r=2)$-LRCs with length $n=3(q+1)$ when $3 \mid (q-1)$ and $q$ is even, and length $n=\frac{3}{2}(q+1)$ when $3 \mid (q-1)$ and $q \equiv 1(\bmod~4)$, respectively. To the best of our knowledge, this is the first construction of $q$-ary cyclic Singleton-optimal LRCs with length $n>q+1$ and minimum distance $d \geq 5$. On the other hand, an LRC acheiving the Hamming-type bound is called a perfect LRC. By using cyclic and constacyclic codes, we construct two new families of $q$-ary perfect LRCs with length $n=\frac{q^m-1}{q-1}$, minimum distance $d=5$ and locality $r=2$.

cs.IT↗