arXiv subjects

Han Mao Kiah

Publications and source records attributed to Han Mao Kiah.

At least 55 records · Page 3Linked to original sources

arXiv2020

Explicit Baranyai Partitions for Quadruples, Part I: Quadrupling Constructions

It is well known that, whenever $k$ divides $n$, the complete $k$-uniform hypergraph on $n$ vertices can be partitioned into disjoint perfect matchings. Equivalently, the set of $k$-subsets of an $n$-set can be partitioned into parallel classes so that each parallel class is a partition of the $n$-set. This result is known as Baranyai's theorem, which guarantees the existence of \emph{Baranyai partitions}. Unfortunately, the proof of Baranyai's theorem uses network flow arguments, making this result non-explicit. In particular, there is no known method to produce Baranyai partitions in time and space that scale linearly with the number of hyperedges in the hypergraph. It is desirable for certain applications to have an explicit construction that generates Baranyai partitions in linear time. Such an efficient construction is known for $k=2$ and $k=3$. In this paper, we present an explicit recursive quadrupling construction for $k=4$ and $n=4t$, where $t \equiv 0,3,4,6,8,9 ~(\text{mod}~12)$. In a follow-up paper (Part II), the other values of~$t$, namely $t \equiv 1,2,5,7,10,11 ~(\text{mod}~12)$, will be considered.

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arXiv2020

Coding for Sequence Reconstruction for Single Edits

The sequence reconstruction problem, introduced by Levenshtein in 2001, considers a communication scenario where the sender transmits a codeword from some codebook and the receiver obtains multiple noisy reads of the codeword. The common setup assumes the codebook to be the entire space and the problem is to determine the minimum number of distinct reads that is required to reconstruct the transmitted codeword. Motivated by modern storage devices, we study a variant of the problem where the number of noisy reads $N$ is fixed. Specifically, we design reconstruction codes that reconstruct a codeword from $N$ distinct noisy reads. We focus on channels that introduce single edit error (i.e. a single substitution, insertion, or deletion) and their variants, and design reconstruction codes for all values of $N$. In particular, for the case of a single edit, we show that as the number of noisy reads increases, the number of redundant bits required can be gracefully reduced from $\log n+O(1)$ to $\log \log n+O(1)$, and then to $O(1)$, where $n$ denotes the length of a codeword. We also show that the redundancy of certain reconstruction codes is within one bit of optimality.

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arXiv2020

Constrained de Bruijn Codes: Properties, Enumeration, Constructions, and Applications

The de Bruijn graph, its sequences, and their various generalizations, have found many applications in information theory, including many new ones in the last decade. In this paper, motivated by a coding problem for emerging memory technologies, a set of sequences which generalize sequences in the de Bruijn graph are defined. These sequences can be also defined and viewed as constrained sequences. Hence, they will be called constrained de Bruijn sequences and a set of such sequences will be called a constrained de Bruijn code. Several properties and alternative definitions for such codes are examined and they are analyzed as generalized sequences in the de Bruijn graph (and its generalization) and as constrained sequences. Various enumeration techniques are used to compute the total number of sequences for any given set of parameters. A construction method of such codes from the theory of shift-register sequences is proposed. Finally, we show how these constrained de Bruijn sequences and codes can be applied in constructions of codes for correcting synchronization errors in the $\ell$-symbol read channel and in the racetrack memory channel. For this purpose, these codes are superior in their size on previously known codes.

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arXiv2020

Repairing Reed-Solomon Codes With Multiple Erasures

Despite their exceptional error-correcting properties, Reed-Solomon codes have been overlooked in distributed storage applications due to the common belief that they have poor repair bandwidth: A naive repair approach would require the whole file to be reconstructed in order to recover a single erased codeword symbol. In a recent work, Guruswami and Wootters (STOC'16) proposed a single-erasure repair method for Reed-Solomon codes that achieves the optimal repair bandwidth amongst all linear encoding schemes. Their key idea is to recover the erased symbol by collecting a sufficiently large number of its traces, each of which can be constructed from a number of traces of other symbols. We extend the trace collection technique to cope with two and three erasures.

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arXiv2020

Optimal Reconstruction Codes for Deletion Channels

The sequence reconstruction problem, introduced by Levenshtein in 2001, considers a communication scenario where the sender transmits a codeword from some codebook and the receiver obtains multiple noisy reads of the codeword. Motivated by modern storage devices, we introduced a variant of the problem where the number of noisy reads $N$ is fixed (Kiah et al. 2020). Of significance, for the single-deletion channel, using $\log_2\log_2 n +O(1)$ redundant bits, we designed a reconstruction code of length $n$ that reconstructs codewords from two distinct noisy reads. In this work, we show that $\log_2\log_2 n -O(1)$ redundant bits are necessary for such reconstruction codes, thereby, demonstrating the optimality of our previous construction. Furthermore, we show that these reconstruction codes can be used in $t$-deletion channels (with $t\ge 2$) to uniquely reconstruct codewords from $n^{t-1}+O\left(n^{t-2}\right)$ distinct noisy reads.

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arXiv2020

Capacity-Approaching Constrained Codes with Error Correction for DNA-Based Data Storage

We propose coding techniques that limit the length of homopolymers runs, ensure the GC-content constraint, and are capable of correcting a single edit error in strands of nucleotides in DNA-based data storage systems. In particular, for given $\ell, ε > 0$, we propose simple and efficient encoders/decoders that transform binary sequences into DNA base sequences (codewords), namely sequences of the symbols A, T, C and G, that satisfy the following properties: (i) Runlength constraint: the maximum homopolymer run in each codeword is at most $\ell$, (ii) GC-content constraint: the GC-content of each codeword is within $[0.5-ε, 0.5+ε]$, (iii) Error-correction: each codeword is capable of correcting a single deletion, or single insertion, or single substitution error. For practical values of $\ell$ and $ε$, we show that our encoders achieve much higher rates than existing results in the literature and approach the capacity. Our methods have low encoding/decoding complexity and limited error propagation.

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arXiv2019

Robust Positioning Patterns with Low Redundancy

A robust positioning pattern is a large array that allows a mobile device to locate its position by reading a possibly corrupted small window around it. In this paper, we provide constructions of binary positioning patterns, equipped with efficient locating algorithms, that are robust to a constant number of errors and have redundancy within a constant factor of optimality. Furthermore, we modify our constructions to correct rank errors and obtain binary positioning patterns robust to any errors of rank less than a constant number. Additionally, we construct $q$-ary robust positioning sequences robust to a large number of errors, some of which have length attaining the upper bound. Our construction of binary positioning sequences that are robust to a constant number of errors has the least known redundancy amongst those explicit constructions with efficient locating algorithms. On the other hand, for binary robust positioning arrays, our construction is the first explicit construction whose redundancy is within a constant factor of optimality. The locating algorithms accompanying both constructions run in time cubic in sequence length or array dimension.

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arXiv2019

Efficient Algorithm for the Linear Complexity of Sequences and Some Related Consequences

The linear complexity of a sequence $s$ is one of the measures of its predictability. It represents the smallest degree of a linear recursion which the sequence satisfies. There are several algorithms to find the linear complexity of a periodic sequence $s$ of length $N$ (where $N$ is of some given form) over a finite field $F_q$ in $O(N)$ symbol field operations. The first such algorithm is The Games-Chan Algorithm which considers binary sequences of period $2^n$, and is known for its extreme simplicity. We generalize this algorithm and apply it efficiently for several families of binary sequences. Our algorithm is very simple, it requires $βN$ bit operations for a small constant $β$, where $N$ is the period of the sequence. We make an analysis on the number of bit operations required by the algorithm and compare it with previous algorithms. In the process, the algorithm also finds the recursion for the shortest linear feedback shift-register which generates the sequence. Some other interesting properties related to shift-register sequences, which might not be too surprising but generally unnoted, are also consequences of our exposition.

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arXiv2019

Optimal Codes Correcting a Single Indel / Edit for DNA-Based Data Storage

An indel refers to a single insertion or deletion, while an edit refers to a single insertion, deletion or substitution. In this paper, we investigate codes that combat either a single indel or a single edit and provide linear-time algorithms that encode binary messages into these codes of length n. Over the quaternary alphabet, we provide two linear-time encoders. One corrects a single edit with log n + O(log log n) redundancy bits, while the other corrects a single indel with log n + 2 redundant bits. These two encoders are order-optimal. The former encoder is the first known order-optimal encoder that corrects a single edit, while the latter encoder (that corrects a single indel) reduces the redundancy of the best known encoder of Tenengolts (1984) by at least four bits. Over the DNA alphabet, we impose an additional constraint: the GC-balanced constraint and require that exactly half of the symbols of any DNA codeword to be either C or G. In particular, via a modification of Knuth's balancing technique, we provide a linear-time map that translates binary messages into GC-balanced codewords and the resulting codebook is able to correct a single indel or a single edit. These are the first known constructions of GC-balanced codes that correct a single indel or a single edit.

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arXiv2019

Proceedings of the 11th Asia-Europe Workshop on Concepts in Information Theory

This year, 2019 we celebrate 30 years of our friendship between Asian and European scientists at the AEW11 in Rotterdam, the Netherlands. Many of the 1989 participants are also present at the 2019 event. This year we have many participants from different parts of Asia and Europe. It shows the importance of this event. It is a good tradition to pay a tribute to a special lecturer in our community. This year we selected Hiroyoshi Morita, who is a well known information theorist with many original contributions.

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arXiv2019

Low-Power Cooling Codes with Efficient Encoding and Decoding

A class of low-power cooling (LPC) codes, to control simultaneously both the peak temperature and the average power consumption of interconnects, was introduced recently. An $(n,t,w)$-LPC code is a coding scheme over $n$ wires that (A) avoids state transitions on the $t$ hottest wires (cooling), and (B) limits the number of transitions to $w$ in each transmission (low-power). A few constructions for large LPC codes that have efficient encoding and decoding schemes, are given. In particular, when $w$ is fixed, we construct LPC codes of size $(n/w)^{w-1}$ and show that these LPC codes can be modified to correct errors efficiently. We further present a construction for large LPC codes based on a mapping from cooling codes to LPC codes. The efficiency of the encoding/decoding for the constructed LPC codes depends on the efficiency of the decoding/encoding for the related cooling codes and the ones for the mapping.

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arXiv2019

Efficient and Explicit Balanced Primer Codes

To equip DNA-based data storage with random-access capabilities, Yazdi et al. (2018) prepended DNA strands with specially chosen address sequences called primers and provided certain design criteria for these primers. We provide explicit constructions of error-correcting codes that are suitable as primer addresses and equip these constructions with efficient encoding algorithms. Specifically, our constructions take cyclic or linear codes as inputs and produce sets of primers with similar error-correcting capabilities. Using certain classes of BCH codes, we obtain infinite families of primer sets of length $n$, minimum distance $d$ with $(d + 1) \log_4 n + O(1)$ redundant symbols. Our techniques involve reversible cyclic codes (1964), an encoding method of Tavares et al. (1971) and Knuth's balancing technique (1986). In our investigation, we also construct efficient and explicit binary balanced error-correcting codes and codes for DNA computing.

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arXiv2019

Generalized Sphere-Packing Bound for Subblock-Constrained Codes

We apply the generalized sphere-packing bound to two classes of subblock-constrained codes. A la Fazeli et al. (2015), we made use of automorphism to significantly reduce the number of variables in the associated linear programming problem. In particular, we study binary constant subblock-composition codes (CSCCs), characterized by the property that the number of ones in each subblock is constant, and binary subblock energy-constrained codes (SECCs), characterized by the property that the number of ones in each subblock exceeds a certain threshold. For CSCCs, we show that the optimization problem is equivalent to finding the minimum of $N$ variables, where $N$ is independent of the number of subblocks. We then provide closed-form solutions for the generalized sphere-packing bounds for single- and double-error correcting CSCCs. For SECCs, we provide closed-form solutions for the generalized sphere-packing bounds for single errors in certain special cases. We also obtain improved bounds on the optimal asymptotic rate for CSCCs and SECCs, and provide numerical examples to highlight the improvement.

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arXiv2018

Domination Mappings into the Hamming Ball: Existence, Constructions, and Algorithms

The Hamming ball of radius $w$ in $\{0,1\}^n$ is the set ${\cal B}(n,w)$ of all binary words of length $n$ and Hamming weight at most $w$. We consider injective mappings $φ: \{0,1\}^m \to {\cal B}(n,w)$ with the following domination property: every position $j \in [n]$ is dominated by some position $i \in [m]$, in the sense that "switching off" position $i$ in $x \in \{0,1\}^m$ necessarily switches off position $j$ in its image $φ(x)$. This property may be described more precisely in terms of a bipartite \emph{domination graph} $G = ([m] \cup [n], E)$ with no isolated vertices, for all $(i,j) \in E$ and all $x \in \{0,1\}^m$, we require that $x_i = 0$ implies $y_j = 0$, where $y = φ(x)$. Although such domination mappings recently found applications in the context of coding for high-performance interconnects, to the best of our knowledge, they were not previously studied. In this paper, we begin with simple necessary conditions for the existence of an $(m,n,w)$-domination mapping $φ: \{0,1\}^m \to {\cal B}(n,w)$. We then provide several explicit constructions of such mappings, which show that the necessary conditions are also sufficient when $w=1$, when $w=2$ and $m$ is odd, or when $m \le 3w$. One of our main results herein is a proof that the trivial necessary condition $|{\cal B}(n,w)| \ge 2^m$ for the existence of an injection is, in fact, sufficient for the existence of an $(m,n,w)$-domination mapping whenever $m$ is sufficiently large. We also present a polynomial-time algorithm that, given any $m$, $n$, and $w$, determines whether an $(m,n,w)$-domination mapping exists for a domination graph with an equitable degree distribution.

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arXiv2018

Efficient Encoding/Decoding of Irreducible Words for Codes Correcting Tandem Duplications

Tandem duplication is the process of inserting a copy of a segment of DNA adjacent to the original position. Motivated by applications that store data in living organisms, Jain et al. (2017) proposed the study of codes that correct tandem duplications. Known code constructions are based on {\em irreducible words}. We study efficient encoding/decoding methods for irreducible words. First, we describe an $(\ell,m)$-finite state encoder and show that when $m=Θ(1/ε)$ and $\ell=Θ(1/ε)$, the encoder achieves rate that is $ε$ away from the optimal. Next, we provide ranking/unranking algorithms for irreducible words and modify the algorithms to reduce the space requirements for the finite state encoder.

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arXiv2017

Deciding the Confusability of Words under Tandem Repeats

Tandem duplication in DNA is the process of inserting a copy of a segment of DNA adjacent to the original position. Motivated by applications that store data in living organisms, Jain {\em et al.} (2016) proposed the study of codes that correct tandem duplications to improve the reliability of data storage. We investigate algorithms associated with the study of these codes. Two words are said to be ${\le}k$-confusable if there exists two sequences of tandem duplications of lengths at most $k$ such that the resulting words are equal. We demonstrate that the problem of deciding whether two words is ${\le}k$-confusable is linear-time solvable through a characterisation that can be checked efficiently for $k=3$. Combining with previous results, the decision problem is linear-time solvable for $k\le 3$. We conjecture that this problem is undecidable for $k>3$. Using insights gained from the algorithm, we study the size of tandem-duplication codes. We improve the previous known upper bound and then construct codes with larger sizes as compared to the previous constructions. We determine the sizes of optimal tandem-duplication codes for lengths up to twenty, develop recursive methods to construct tandem-duplication codes for all word lengths, and compute explicit lower bounds for the size of optimal tandem-duplication codes for lengths from 21 to 30.

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arXiv2017

Cooling Codes: Thermal-Management Coding for High-Performance Interconnects

High temperatures have dramatic negative effects on interconnect performance and, hence, numerous techniques have been proposed to reduce the power consumption of on-chip buses. However, existing methods fall short of fully addressing the thermal challenges posed by high-performance interconnects. In this paper, we introduce new efficient coding schemes that make it possible to directly control the peak temperature of a bus by effectively cooling its hottest wires. This is achieved by avoiding state transitions on the hottest wires for as long as necessary until their temperature drops off. We also reduce the average power consumption by making sure that the total number of state transitions on all the wires is below a prescribed threshold. We show how each of these two features can be coded for separately or, alternatively, how both can be achieved at the same time. In addition, error-correction for the transmitted information can be provided while controlling the peak temperature and/or the average power consumption. In general, our cooling codes use $n > k$ wires to encode a given $k$-bit bus. One of our goals herein is to determine the minimum possible number of wires $n$ needed to encode $k$ bits while satisfying any combination of the three desired properties. We provide full theoretical analysis in each case. In particular, we show that $n = k+t+1$ suffices to cool the $t$ hottest wires, and this is the best possible. Moreover, although the proposed coding schemes make use of sophisticated tools from combinatorics, discrete geometry, linear algebra, and coding theory, the resulting encoders and decoders are fully practical. They do not require significant computational overhead and can be implemented without sacrificing a large circuit area.

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arXiv2017

Mutually Uncorrelated Primers for DNA-Based Data Storage

We introduce the notion of weakly mutually uncorrelated (WMU) sequences, motivated by applications in DNA-based data storage systems and for synchronization of communication devices. WMU sequences are characterized by the property that no sufficiently long suffix of one sequence is the prefix of the same or another sequence. WMU sequences used for primer design in DNA-based data storage systems are also required to be at large mutual Hamming distance from each other, have balanced compositions of symbols, and avoid primer-dimer byproducts. We derive bounds on the size of WMU and various constrained WMU codes and present a number of constructions for balanced, error-correcting, primer-dimer free WMU codes using Dyck paths, prefix-synchronized and cyclic codes.

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