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Yeow Meng Chee

Publications and source records attributed to Yeow Meng Chee.

At least 55 records · Page 3Linked to original sources
arXiv2019

Access Balancing in Storage Systems by Labeling Partial Steiner Systems

Storage architectures ranging from minimum bandwidth regenerating encoded distributed storage systems to declustered-parity RAIDs can be designed using dense partial Steiner systems in order to support fast reads, writes, and recovery of failed storage units. In order to ensure good performance, popularities of the data items should be taken into account and the frequencies of accesses to the storage units made as uniform as possible. A proposed combinatorial model ranks items by popularity and assigns data items to elements in a dense partial Steiner system so that the sums of ranks of the elements in each block are as equal as possible. By developing necessary conditions in terms of independent sets, we demonstrate that certain Steiner systems must have a much larger difference between the largest and smallest block sums than is dictated by an elementary lower bound. In contrast, we also show that certain dense partial $S(t, t+1, v)$ designs can be labeled to realize the elementary lower bound. Furthermore, we prove that for every admissible order $v$, there is a Steiner triple system $(S(2, 3, v))$ whose largest difference in block sums is within an additive constant of the lower bound.

cs.IT↗
arXiv2019

Optimal $q$-Ary Error Correcting/All Unidirectional Error Detecting Codes

Codes that can correct up to $t$ symmetric errors and detect all unidirectional errors, known as $t$-EC-AUED codes, are studied in this paper. Given positive integers $q$, $a$ and $t$, let $n_q(a,t+1)$ denote the length of the shortest $q$-ary $t$-EC-AUED code of size $a$. We introduce combinatorial constructions for $q$-ary $t$-EC-AUED codes via one-factorizations of complete graphs, and concatenation of MDS codes and codes from resolvable set systems. Consequently, we determine the exact values of $n_q(a,t+1)$ for several new infinite families of $q,a$ and $t$.

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arXiv2019

Complexity of Dependencies in Bounded Domains, Armstrong Codes, and Generalizations

The study of Armstrong codes is motivated by the problem of understanding complexities of dependencies in relational database systems, where attributes have bounded domains. A $(q,k,n)$-Armstrong code is a $q$-ary code of length $n$ with minimum Hamming distance $n-k+1$, and for any set of $k-1$ coordinates there exist two codewords that agree exactly there. Let $f(q,k)$ be the maximum $n$ for which such a code exists. In this paper, $f(q,3)=3q-1$ is determined for all $q\geq 5$ with three possible exceptions. This disproves a conjecture of Sali. Further, we introduce generalized Armstrong codes for branching, or $(s,t)$-dependencies, construct several classes of optimal Armstrong codes and establish lower bounds for the maximum length $n$ in this more general setting.

math.CO↗
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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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.

math.CO↗
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.

math.CO↗
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

Coding for Racetrack Memories

Racetrack memory is a new technology which utilizes magnetic domains along a nanoscopic wire in order to obtain extremely high storage density. In racetrack memory, each magnetic domain can store a single bit of information, which can be sensed by a reading port (head). The memory has a tape-like structure which supports a shift operation that moves the domains to be read sequentially by the head. In order to increase the memory's speed, prior work studied how to minimize the latency of the shift operation, while the no less important reliability of this operation has received only a little attention. In this work we design codes which combat shift errors in racetrack memory, called position errors. Namely, shifting the domains is not an error-free operation and the domains may be over-shifted or are not shifted, which can be modeled as deletions and sticky insertions. While it is possible to use conventional deletion and insertion-correcting codes, we tackle this problem with the special structure of racetrack memory, where the domains can be read by multiple heads. Each head outputs a noisy version of the stored data and the multiple outputs are combined in order to reconstruct the data. Under this paradigm, we will show that it is possible to correct, with at most a single bit of redundancy, $d$ deletions with $d+1$ heads if the heads are well-separated. Similar results are provided for burst of deletions, sticky insertions and combinations of both deletions and sticky insertions.

cs.IT↗
arXiv2016

Linear Size Constant-Composition Codes Meeting the Johnson Bound

The Johnson-type upper bound on the maximum size of a code of length $n$, distance $d=2w-1$ and constant composition ${\overline{w}}$ is $\lfloor\dfrac{n}{w_1}\rfloor$, where $w$ is the total weight and $w_1$ is the largest component of ${\overline{w}}$. Recently, Chee et al. proved that this upper bound can be achieved for all constant-composition codes of sufficiently large lengths. Let $N_{ccc}({\overline{w}})$ be the smallest such length. The determination of $N_{ccc}({\overline{w}})$ is trivial for binary codes. This paper provides a lower bound on $N_{ccc}({\overline{w}})$, which is shown to be tight for all ternary and quaternary codes by giving new combinatorial constructions. Consequently, by refining method, we determine the values of $N_{ccc}({\overline{w}})$ for all $q$-ary constant-composition codes provided that $3w_1\geq w$ with finite possible exceptions.

math.CO↗
arXiv2015

Product Construction of Affine Codes

Binary matrix codes with restricted row and column weights are a desirable method of coded modulation for power line communication. In this work, we construct such matrix codes that are obtained as products of affine codes - cosets of binary linear codes. Additionally, the constructions have the property that they are systematic. Subsequently, we generalize our construction to irregular product of affine codes, where the component codes are affine codes of different rates.

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arXiv2014

Constructions of Optimal and Near-Optimal Multiply Constant-Weight Codes

Multiply constant-weight codes (MCWCs) have been recently studied to improve the reliability of certain physically unclonable function response. In this paper, we give combinatorial constructions for MCWCs which yield several new infinite families of optimal MCWCs. Furthermore, we demonstrate that the Johnson type upper bounds of MCWCs are asymptotically tight for fixed weights and distances. Finally, we provide bounds and constructions of two dimensional MCWCs.

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arXiv2014

Generalized Balanced Tournament Packings and Optimal Equitable Symbol Weight Codes for Power Line Communications

Generalized balance tournament packings (GBTPs) extend the concept of generalized balanced tournament designs introduced by Lamken and Vanstone (1989). In this paper, we establish the connection between GBTPs and a class of codes called equitable symbol weight codes. The latter were recently demonstrated to optimize the performance against narrowband noise in a general coded modulation scheme for power line communications. By constructing classes of GBTPs, we establish infinite families of optimal equitable symbol weight codes with code lengths greater than alphabet size and whose narrowband noise error-correcting capability to code length ratios do not diminish to zero as the length grows.

math.CO↗
arXiv2014

Multiply Constant-Weight Codes and the Reliability of Loop Physically Unclonable Functions

We introduce the class of multiply constant-weight codes to improve the reliability of certain physically unclonable function (PUF) response. We extend classical coding methods to construct multiply constant-weight codes from known $q$-ary and constant-weight codes. Analogues of Johnson bounds are derived and are shown to be asymptotically tight to a constant factor under certain conditions. We also examine the rates of the multiply constant-weight codes and interestingly, demonstrate that these rates are the same as those of constant-weight codes of suitable parameters. Asymptotic analysis of our code constructions is provided.

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arXiv2013

Importance of Symbol Equity in Coded Modulation for Power Line Communications

The use of multiple frequency shift keying modulation with permutation codes addresses the problem of permanent narrowband noise disturbance in a power line communications system. In this paper, we extend this coded modulation scheme based on permutation codes to general codes and introduce an additional new parameter that more precisely captures a code's performance against permanent narrowband noise. As a result, we define a new class of codes, namely, equitable symbol weight codes, which are optimal with respect to this measure.

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arXiv2013

Maximum Distance Separable Codes for Symbol-Pair Read Channels

We study (symbol-pair) codes for symbol-pair read channels introduced recently by Cassuto and Blaum (2010). A Singleton-type bound on symbol-pair codes is established and infinite families of optimal symbol-pair codes are constructed. These codes are maximum distance separable (MDS) in the sense that they meet the Singleton-type bound. In contrast to classical codes, where all known q-ary MDS codes have length O(q), we show that q-ary MDS symbol-pair codes can have length Ω(q^2). In addition, we completely determine the existence of MDS symbol-pair codes for certain parameters.

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