arXiv subjects

Han Mao Kiah

Publications and source records attributed to Han Mao Kiah.

At least 19 recordsLinked to original sources

arXiv2026

Constant-List Insertion--Deletion Codes:New Bounds and an Improvement of Levenshtein's Lower Bound

We study codes correcting adversarial insertions and deletions with list size $L$ fixed independently of the block length. We derive new achievable-rate bounds for binary codes and upper bounds over every fixed alphabet of size $q\ge2$, retaining explicit dependence on $L$. We establish a combinatorial reduction that trades $L$ units of insertion budget for one unit of deletion budget in the decoding guarantee, without changing the code or increasing the list size. Consequently, asymptotic bounds for mixed errors with insertion fraction $γ$ and deletion fraction $δ$ follow from insertion-only lower bounds at $γ+Lδ$ and deletion-only upper bounds at $δ+γ/L$. For binary unique decoding, we strictly improve Levenshtein's classical asymptotic rate lower bound for every deletion fraction $0<δ<1/2$ for which the classical rate expression is nonnegative. At $δ=0.1$, the lower bound increases from approximately $0.162009$ to $0.180431$, a relative increase of about $11.37\%$. Our framework also yields insertion and deletion lower bounds for every fixed list size. The existence proofs combine the Lovász local lemma with sampling from words having a specified number of runs, where a run is a maximal block of equal symbols. Generating functions provide refined bounds on the probability that $L+1$ sampled words share an allowed received word. We also derive a Levenshtein-type upper bound by run counting and, separately, a higher-order Elias bound using intersections and unions of the position sets used to embed $L+1$ codewords in a common supersequence. The latter recovers Yasunaga's asymptotic unique-decoding bound at $L=1$ and strictly improves the Haeupler--Shahrasbi--Sudan insertion bound for every fixed $L$ and $0<γ<q-1$. Numerical comparisons quantify the gains and remaining gaps.

cs.IT↗

arXiv2026

Bounds on Codes Correcting Adjacent Transpositions

We study the problem of correcting pairwise disjoint adjacent transpositions (or swaps) in $q$-ary strings. Equivalently, the model we assume is the radius-one instance of the so-called $\ell_\infty$-limited permutation channel. We first study the relevant combinatorial properties of the appropriately defined transposition distance, including center-specific and average ball sizes. We then derive two lower bounds and one upper bound on the asymptotic rates of optimal codes correcting $t=τn$ transpositions. The first achievability result is a generalized Gilbert--Varshamov bound, while the second follows from a construction of codes correcting all possible patterns of adjacent transpositions and therefore represents a lower bound on the zero-error capacity of this model. This construction improves the classical general-alphabet construction for $3\leqslant q\leqslant 8$ as well as the recent bounds for $q=4,5$. The upper bound is obtained by a packing argument adjusted to the run-structure of a given code. To the best of our knowledge, these are the first nonconstant, $τ$-dependent lower and upper bounds developed for the pairwise disjoint $q$-ary model throughout the linear regime. We also derive asymptotic bounds on the cardinality of optimal codes correcting $t=\textrm{const}$ pairwise disjoint adjacent transpositions.

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↗

arXiv2026

Expected Recovery Time in DNA-based Distributed Storage Systems

We initiate the study of DNA-based distributed storage systems, where information is encoded across multiple DNA data storage containers to achieve robustness against container failures. In this setting, data are distributed over $M$ containers, and the objective is to guarantee that the contents of any failed container can be reliably reconstructed from the surviving ones. Unlike classical distributed storage systems, DNA data storage containers are fundamentally constrained by sequencing technology, since each read operation yields the content of a uniformly random sampled strand from the container. Within this framework, we consider several erasure-correcting codes and analyze the expected recovery time of the data stored in a failed container. Our results are obtained by analyzing generalized versions of the classical Coupon Collector's Problem, which may be of independent interest.

cs.IT↗

arXiv2026

Reconstructing Reed-Solomon Codes from Multiple Noisy Channel Outputs

The sequence reconstruction problem, introduced by Levenshtein in 2001, considers a communication setting in which a sender transmits a codeword and the receiver observes K independent noisy versions of this codeword. In this work, we study the problem of efficient reconstruction when each of the $K$ outputs is corrupted by a $q$-ary discrete memoryless symmetric (DMS) substitution channel with substitution probability $p$. Focusing on Reed-Solomon (RS) codes, we adapt the Koetter-Vardy soft-decision decoding algorithm to obtain an efficient reconstruction algorithm. For sufficiently large blocklength and alphabet size, we derive an explicit rate threshold, depending only on $(p, K)$, such that the transmitted codeword can be reconstructed with arbitrarily small probability of error whenever the code rate $R$ lies below this threshold.

cs.IT↗

arXiv2026

Trace Repair Never Loses to Classical Repair: Exact and Explicit Helper Nodes Selection

Repairing Reed-Solomon codes with low bandwidth is a central challenge in distributed storage. Following the trace-repair framework of Guruswami and Wootters (2017), recent works by Lin (2023) and Liu-Wan-Xing (2024) provided significant improvements in bandwidth using two distinct ideas. Lin constructed a trace-repair scheme that requires no contribution from a set of predetermined nodes $\mathscr{S}$, while Liu-Wan-Xing identified linear dependencies among the downloaded traces, relating the number of dependent traces to the dimension of a subspace $\mathscr{W}_k$. In this work, we fully utilize and unify these ideas. We compute the exact dimension of $\mathscr{W}_{k,\mathscr{S}}$ (a generalization of $\mathscr{W}_k$). We identify the trade-off between the set size $|\mathscr{S}|$ and the dimension $\dim(\mathscr{W}_{k,\mathscr{S}})$. We provide an algorithm to find the combination that results in the lowest bandwidth. Furthermore, we provide an explicit choice of the helper nodes for the repair. Finally, we prove that our optimized scheme never loses to the classical repair scheme, establishing a bandwidth guarantee of at most $k\log|\mathbb{F}|$ bits for all dimension $k$ and field $\mathbb{F}$, whenever the trace repair is applicable.

cs.IT↗

arXiv2025

Zigzag Codes Revisited: From Optimal Rebuilding to Small Skip Cost and Small Fields

We revisit zigzag array codes, a family of MDS codes known for achieving optimal access and optimal rebuilding ratio in single-node repair. In this work, we endow zigzag codes with two new properties: small field size and low skip cost. First, we prove that when the row-indexing group is $\mathcal{G} = \mathbb{Z}_2^m$ and the field has characteristic two, explicit coefficients over any field with $|\mathcal{F}|\ge N$ guarantee the MDS property, thereby decoupling the dependence among $p$, $k$, and $M$. Second, we introduce an ordering-and-subgroup framework that yields repair-by-transfer schemes with bounded skip cost and low repair-fragmentation ratio (RFR), while preserving optimal access and optimal rebuilding ratio. Our explicit constructions include families with zero skip cost whose rates approach $2/3$, and families with bounded skip cost whose rates approach $3/4$ and $4/5$. These rates are comparable to those of MDS array codes widely deployed in practice. Together, these results demonstrate that zigzag codes can be made both more flexible in theory and more practical for modern distributed storage systems.

cs.IT↗

arXiv2025

BMTree: Designing, Learning, and Updating Piecewise Space-Filling Curves for Multi-Dimensional Data Indexing

Space-filling curves (SFC, for short) have been widely applied to index multi-dimensional data, which first maps the data to one dimension, and then a one-dimensional indexing method, e.g., the B-tree indexes the mapped data. Existing SFCs adopt a single mapping scheme for the whole data space. However, a single mapping scheme often does not perform well on all the data space. In this paper, we propose a new type of SFC called piecewise SFCs that adopts different mapping schemes for different data subspaces. Specifically, we propose a data structure termed the Bit Merging tree (BMTree) that can generate data subspaces and their SFCs simultaneously, and achieve desirable properties of the SFC for the whole data space. Furthermore, we develop a reinforcement learning-based solution to build the BMTree, aiming to achieve excellent query performance. To update the BMTree efficiently when the distributions of data and/or queries change, we develop a new mechanism that achieves fast detection of distribution shifts in data and queries, and enables partial retraining of the BMTree. The retraining mechanism achieves performance enhancement efficiently since it avoids retraining the BMTree from scratch. Extensive experiments show the effectiveness and efficiency of the BMTree with the proposed learning-based methods.

cs.DB↗

arXiv2024

Pairs in Nested Steiner Quadruple Systems

Motivated by a repair problem for fractional repetition codes in distributed storage, each block of any Steiner quadruple system (SQS) of order $v$ is partitioned into two pairs. Each pair in such a partition is called a nested design pair and its multiplicity is the number of times it is a pair in this partition. Such a partition of each block is considered as a new block design called a nested Steiner quadruple system. Several related questions on this type of design are considered in this paper: What is the maximum multiplicity of the nested design pair with minimum multiplicity? What is the minimum multiplicity of the nested design pair with maximum multiplicity? Are there nested quadruple systems in which all the nested design pairs have the same multiplicity? Of special interest are nested quadruple systems in which all the $\binom{v}{2}$ pairs are nested design pairs with the same multiplicity. Several constructions of nested quadruple systems are considered and in particular classic constructions of SQS are examined.

math.CO↗

arXiv2024

Repairing Reed-Solomon Codes with Side Information

We generalize the problem of recovering a lost/erased symbol in a Reed-Solomon code to the scenario in which some side information about the lost symbol is known. The side information is represented as a set $S$ of linearly independent combinations of the sub-symbols of the lost symbol. When $S = \varnothing$, this reduces to the standard problem of repairing a single codeword symbol. When $S$ is a set of sub-symbols of the erased one, this becomes the repair problem with partially lost/erased symbol. We first establish that the minimum repair bandwidth depends on $|S|$ and not the content of $S$ and construct a lower bound on the repair bandwidth of a linear repair scheme with side information $S$. We then consider the well-known subspace-polynomial repair schemes and show that their repair bandwidths can be optimized by choosing the right subspaces. Finally, we demonstrate several parameter regimes where the optimal bandwidths can be achieved for full-length Reed-Solomon codes.

cs.IT↗

arXiv2024

Noise-Tolerant Codebooks for Semi-Quantitative Group Testing: Application to Spatial Genomics

Motivated by applications in spatial genomics, we revisit group testing (Dorfman~1943) and propose the class of $λ$-{\sf ADD}-codes, studying such codes with certain distance $d$ and codelength $n$. When $d$ is constant, we provide explicit code constructions with rates close to $1/2$. When $d$ is proportional to $n$, we provide a GV-type lower bound whose rates are efficiently computable. Upper bounds for such codes are also studied.

cs.IT↗

arXiv2024

Private Repair of a Single Erasure in Reed-Solomon Codes

We investigate the problem of privately recovering a single erasure for Reed-Solomon codes with low communication bandwidths. For an $[n,k]_{q^\ell}$ code with $n-k\geq q^{m}+t-1$, we construct a repair scheme that allows a client to recover an arbitrary codeword symbol without leaking its index to any set of $t$ colluding helper nodes at a repair bandwidth of $(n-1)(\ell-m)$ sub-symbols in $\mathbb{F}_q$. When $t=1$, this reduces to the bandwidth of existing repair schemes based on subspace polynomials. We prove the optimality of the proposed scheme when $n=q^\ell$ under a reasonable assumption about the schemes being used. Our private repair scheme can also be transformed into a private retrieval scheme for data encoded by Reed-Solomon codes.

cs.IT↗

arXiv2024

Repairing with Zero Skip Cost

To measure repair latency at helper nodes, we introduce a new metric called skip cost that quantifies the number of contiguous sections accessed on a disk. We provide explicit constructions of zigzag codes and fractional repetition codes that incur zero skip cost

cs.IT↗

arXiv2024

Coding for Synthesis Defects

Motivated by DNA based data storage system, we investigate the errors that occur when synthesizing DNA strands in parallel, where each strand is appended one nucleotide at a time by the machine according to a template supersequence. If there is a cycle such that the machine fails, then the strands meant to be appended at this cycle will not be appended, and we refer to this as a synthesis defect. In this paper, we present two families of codes correcting synthesis defects, which are t-known-synthesis-defect correcting codes and t-synthesis-defect correcting codes. For the first one, it is assumed that the defective cycles are known, and each of the codeword is a quaternary sequence. We provide constructions for this family of codes for t = 1, 2, with redundancy log 4 and log n+18 log 3, respectively. For the second one, the codeword is a set of M ordered sequences, and we give constructions for t = 1, 2 to show a strategy for constructing this family of codes. Finally, we derive a lower bound on the redundancy for single-known-synthesis-defect correcting codes, which assures that our construction is almost optimal.

cs.IT↗

arXiv2024

Recovery Sets of Subspaces from a Simplex Code

Recovery sets for vectors and subspaces are important in the construction of distributed storage system codes. These concepts are also interesting in their own right. In this paper, we consider the following very basic recovery question: what is the maximum number of possible pairwise disjoint recovery sets for each recovered element? The recovered elements in this work are d-dimensional subspaces of a $k$-dimensional vector space over GF(q). Each server stores one representative for each distinct one-dimensional subspace of the k-dimensional vector space, or equivalently a distinct point of PG(k-1,q). As column vectors, the associated vectors of the stored one-dimensional subspaces form the generator matrix of the $[(q^k -1)/(q-1),k,q^{k-1}]$ simplex code over GF(q). Lower bounds and upper bounds on the maximum number of such recovery sets are provided. It is shown that generally, these bounds are either tight or very close to being tight.

cs.IT↗

arXiv2024

Permutation Recovery Problem against Deletion Errors for DNA Data Storage

Owing to its immense storage density and durability, DNA has emerged as a promising storage medium. However, due to technological constraints, data can only be written onto many short DNA molecules called data blocks that are stored in an unordered way. To handle the unordered nature of DNA data storage systems, a unique address is typically prepended to each data block to form a DNA strand. However, DNA storage systems are prone to errors and generate multiple noisy copies of each strand called DNA reads. Thus, we study the permutation recovery problem against deletions errors for DNA data storage. The permutation recovery problem for DNA data storage requires one to reconstruct the addresses or in other words to uniquely identify the noisy reads. By successfully reconstructing the addresses, one can essentially determine the correct order of the data blocks, effectively solving the clustering problem. We first show that we can almost surely identify all the noisy reads under certain mild assumptions. We then propose a permutation recovery procedure and analyze its complexity.

cs.IT↗

arXiv2024

An Optimal Sequence Reconstruction Algorithm for Reed-Solomon Codes

The sequence reconstruction problem, introduced by Levenshtein in 2001, considers a scenario where the sender transmits a codeword from some codebook, and the receiver obtains $N$ noisy outputs of the codeword. We study the problem of efficient reconstruction using $N$ outputs that are each corrupted by at most $t$ substitutions. Specifically, for the ubiquitous Reed-Solomon codes, we adapt the Koetter-Vardy soft-decoding algorithm, presenting a reconstruction algorithm capable of correcting beyond Johnson radius. Furthermore, the algorithm uses $\mathcal{O}(nN)$ field operations, where $n$ is the codeword length.

cs.IT↗

arXiv2024

Evaluating the Gilbert-Varshamov Bound for Constrained Systems

We revisit the well-known Gilbert-Varshamov (GV) bound for constrained systems. In 1991, Kolesnik and Krachkovsky showed that GV bound can be determined via the solution of some optimization problem. Later, Marcus and Roth (1992) modified the optimization problem and improved the GV bound in many instances. In this work, we provide explicit numerical procedures to solve these two optimization problems and hence, compute the bounds. We then show the procedures can be further simplified when we plot the respective curves. In the case where the graph presentation comprise a single state, we provide explicit formulas for both bounds.

cs.IT↗