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Yeow Meng Chee

Publications and source records attributed to Yeow Meng Chee.

At least 73 records · Page 4Linked to original sources
arXiv2013

Polynomial Time Algorithm for Min-Ranks of Graphs with Simple Tree Structures

The min-rank of a graph was introduced by Haemers (1978) to bound the Shannon capacity of a graph. This parameter of a graph has recently gained much more attention from the research community after the work of Bar-Yossef et al. (2006). In their paper, it was shown that the min-rank of a graph G characterizes the optimal scalar linear solution of an instance of the Index Coding with Side Information (ICSI) problem described by the graph G. It was shown by Peeters (1996) that computing the min-rank of a general graph is an NP-hard problem. There are very few known families of graphs whose min-ranks can be found in polynomial time. In this work, we introduce a new family of graphs with efficiently computed min-ranks. Specifically, we establish a polynomial time dynamic programming algorithm to compute the min-ranks of graphs having simple tree structures. Intuitively, such graphs are obtained by gluing together, in a tree-like structure, any set of graphs for which the min-ranks can be determined in polynomial time. A polynomial time algorithm to recognize such graphs is also proposed.

math.CO↗
arXiv2013

Upper Bounds on Matching Families in $\mathbb{Z}_{pq}^n$

\textit{Matching families} are one of the major ingredients in the construction of {\em locally decodable codes} (LDCs) and the best known constructions of LDCs with a constant number of queries are based on matching families. The determination of the largest size of any matching family in $\mathbb{Z}_m^n$, where $\mathbb{Z}_m$ is the ring of integers modulo $m$, is an interesting problem. In this paper, we show an upper bound of $O((pq)^{0.625n+0.125})$ for the size of any matching family in $\mathbb{Z}_{pq}^n$, where $p$ and $q$ are two distinct primes. Our bound is valid when $n$ is a constant, $p\rightarrow \infty$ and $p/q\rightarrow 1$. Our result improves an upper bound of Dvir {\it et al.}

cs.IT↗
arXiv2013

Optimal Index Codes with Near-Extreme Rates

The min-rank of a digraph was shown by Bar-Yossef et al. (2006) to represent the length of an optimal scalar linear solution of the corresponding instance of the Index Coding with Side Information (ICSI) problem. In this work, the graphs and digraphs of near-extreme min-ranks are characterized. Those graphs and digraphs correspond to the ICSI instances having near-extreme transmission rates when using optimal scalar linear index codes. In particular, it is shown that the decision problem whether a digraph has min-rank two is NP-complete. By contrast, the same question for graphs can be answered in polynomial time. Additionally, a new upper bound on the min-rank of a digraph, the circuit-packing bound, is presented. This bound is often tighter than the previously known bounds. By employing this new bound, we present several families of digraphs whose min-ranks can be found in polynomial time.

cs.IT↗
arXiv2013

Cross-Bifix-Free Codes Within a Constant Factor of Optimality

A cross-bifix-free code is a set of words in which no prefix of any length of any word is the suffix of any word in the set. Cross-bifix-free codes arise in the study of distributed sequences for frame synchronization. We provide a new construction of cross-bifix-free codes which generalizes the construction in Bajic (2007) to longer code lengths and to any alphabet size. The codes are shown to be nearly optimal in size. We also establish new results on Fibonacci sequences, that are used in estimating the size of the cross-bifix-free codes.

cs.IT↗
arXiv2012

Estimates on the Size of Symbol Weight Codes

The study of codes for powerlines communication has garnered much interest over the past decade. Various types of codes such as permutation codes, frequency permutation arrays, and constant composition codes have been proposed over the years. In this work we study a type of code called the bounded symbol weight codes which was first introduced by Versfeld et al. in 2005, and a related family of codes that we term constant symbol weight codes. We provide new upper and lower bounds on the size of bounded symbol weight and constant symbol weight codes. We also give direct and recursive constructions of codes for certain parameters.

cs.IT↗
arXiv2012

On Bringer-Chabanne EPIR Protocol for Polynomial Evaluation

Extended private information retrieval (EPIR) was defined by \cite{BCPT07} at CANS'07 and generalized by \cite{BC09} at AFRICACRYPT'09. In the generalized setting, EPIR allows a user to evaluate a function on a database block such that the database can learn neither which function has been evaluated nor on which block the function has been evaluated and the user learns no more information on the database blocks except for the expected result. An EPIR protocol for evaluating polynomials over a finite field $L$ was proposed by Bringer and Chabanne in \cite{BC09}. We show that the protocol does not satisfy the correctness requirement as they have claimed. In particular, we show that it does not give the user the expected result with large probability if one of the coefficients of the polynomial to be evaluated is primitive in $L$ and the others belong to the prime subfield of $L$.

cs.CR↗
arXiv2012

Improved Constructions of Frameproof Codes

Frameproof codes are used to preserve the security in the context of coalition when fingerprinting digital data. Let $M_{c,l}(q)$ be the largest cardinality of a $q$-ary $c$-frameproof code of length $l$ and $R_{c,l}=\lim_{q\rightarrow \infty}M_{c,l}(q)/q^{\lceil l/c\rceil}$. It has been determined by Blackburn that $R_{c,l}=1$ when $l\equiv 1\ (\bmod\ c)$, $R_{c,l}=2$ when $c=2$ and $l$ is even, and $R_{3,5}=5/3$. In this paper, we give a recursive construction for $c$-frameproof codes of length $l$ with respect to the alphabet size $q$. As applications of this construction, we establish the existence results for $q$-ary $c$-frameproof codes of length $c+2$ and size $\frac{c+2}{c}(q-1)^2+1$ for all odd $q$ when $c=2$ and for all $q\equiv 4\pmod{6}$ when $c=3$. Furthermore, we show that $R_{c,c+2}=(c+2)/c$ meeting the upper bound given by Blackburn, for all integers $c$ such that $c+1$ is a prime power.

math.CO↗
arXiv2011

Error Correction for Index Coding with Side Information

A problem of index coding with side information was first considered by Y. Birk and T. Kol (IEEE INFOCOM, 1998). In the present work, a generalization of index coding scheme, where transmitted symbols are subject to errors, is studied. Error-correcting methods for such a scheme, and their parameters, are investigated. In particular, the following question is discussed: given the side information hypergraph of index coding scheme and the maximal number of erroneous symbols $δ$, what is the shortest length of a linear index code, such that every receiver is able to recover the required information? This question turns out to be a generalization of the problem of finding a shortest-length error-correcting code with a prescribed error-correcting capability in the classical coding theory. The Singleton bound and two other bounds, referred to as the $α$-bound and the $κ$-bound, for the optimal length of a linear error-correcting index code (ECIC) are established. For large alphabets, a construction based on concatenation of an optimal index code with an MDS classical code, is shown to attain the Singleton bound. For smaller alphabets, however, this construction may not be optimal. A random construction is also analyzed. It yields another implicit bound on the length of an optimal linear ECIC. Further, the problem of error-correcting decoding by a linear ECIC is studied. It is shown that in order to decode correctly the desired symbol, the decoder is required to find one of the vectors, belonging to an affine space containing the actual error vector. The syndrome decoding is shown to produce the correct output if the weight of the error pattern is less or equal to the error-correcting capability of the corresponding ECIC. Finally, the notion of static ECIC, which is suitable for use with a family of instances of an index coding problem, is introduced.

cs.IT↗
arXiv2011

On the Security of Index Coding with Side Information

Security aspects of the Index Coding with Side Information (ICSI) problem are investigated. Building on the results of Bar-Yossef et al. (2006), the properties of linear index codes are further explored. The notion of weak security, considered by Bhattad and Narayanan (2005) in the context of network coding, is generalized to block security. It is shown that the linear index code based on a matrix $L$, whose column space code $C(L)$ has length $n$, minimum distance $d$ and dual distance $d^\perp$, is $(d-1-t)$-block secure (and hence also weakly secure) if the adversary knows in advance $t \leq d-2$ messages, and is completely insecure if the adversary knows in advance more than $n - d$ messages. Strong security is examined under the conditions that the adversary: (i) possesses $t$ messages in advance; (ii) eavesdrops at most $μ$ transmissions; (iii) corrupts at most $δ$ transmissions. We prove that for sufficiently large $q$, an optimal linear index code which is strongly secure against such an adversary has length $κ_q+μ+2δ$. Here $κ_q$ is a generalization of the min-rank over $F_q$ of the side information graph for the ICSI problem in its original formulation in the work of Bar- Yossef et al.

cs.IT↗
arXiv2011

Index Coding and Error Correction

A problem of index coding with side information was first considered by Y. Birk and T. Kol (IEEE INFOCOM, 1998). In the present work, a generalization of index coding scheme, where transmitted symbols are subject to errors, is studied. Error-correcting methods for such a scheme, and their parameters, are investigated. In particular, the following question is discussed: given the side information hypergraph of index coding scheme and the maximal number of erroneous symbols $δ$, what is the shortest length of a linear index code, such that every receiver is able to recover the required information? This question turns out to be a generalization of the problem of finding a shortest-length error-correcting code with a prescribed error-correcting capability in the classical coding theory. The Singleton bound and two other bounds, referred to as the $α$-bound and the $κ$-bound, for the optimal length of a linear error-correcting index code (ECIC) are established. For large alphabets, a construction based on concatenation of an optimal index code with an MDS classical code, is shown to attain the Singleton bound. For smaller alphabets, however, this construction may not be optimal. A random construction is also analyzed. It yields another inexplicit bound on the length of an optimal linear ECIC. Finally, the decoding of linear ECIC's is discussed. The syndrome decoding is shown to output the exact message if the weight of the error vector is less or equal to the error-correcting capability of the corresponding ECIC.

cs.IT↗
arXiv2010

Secure Index Coding with Side Information

Security aspects of the Index Coding with Side Information (ICSI) problem are investigated. Building on the results of Bar-Yossef et al. (2006), the properties of linear coding schemes for the ICSI problem are further explored. The notion of weak security, considered by Bhattad and Narayanan (2005) in the context of network coding, is generalized to block security. It is shown that the coding scheme for the ICSI problem based on a linear code C of length n, minimum distance d and dual distance d^\perp, is (d-1-t)-block secure (and hence also weakly secure) if the adversary knows in advance t \le d - 2 messages, and is completely insecure if the adversary knows in advance more than n - d^\perp messages.

cs.IT↗
arXiv2010

Strongly Regular Graphs Constructed from $p$-ary Bent Functions

In this paper, we generalize the construction of strongly regular graphs in [Y. Tan et al., Strongly regular graphs associated with ternary bent functions, J. Combin.Theory Ser. A (2010), 117, 668-682] from ternary bent functions to $p$-ary bent functions, where $p$ is an odd prime. We obtain strongly regular graphs with three types of parameters. Using certain non-quadratic $p$-ary bent functions, our constructions can give rise to new strongly regular graphs for small parameters.

math.CO↗
arXiv2010

List Decodability at Small Radii

$A'(n,d,e)$, the smallest $\ell$ for which every binary error-correcting code of length $n$ and minimum distance $d$ is decodable with a list of size $\ell$ up to radius $e$, is determined for all $d\geq 2e-3$. As a result, $A'(n,d,e)$ is determined for all $e\leq 4$, except for 42 values of $n$.

cs.IT↗
arXiv2010

A Pair of Disjoint 3-GDDs of type g^t u^1

Pairwise disjoint 3-GDDs can be used to construct some optimal constant-weight codes. We study the existence of a pair of disjoint 3-GDDs of type $g^t u^1$ and establish that its necessary conditions are also sufficient.

math.CO↗
arXiv2010

Universal Cycles for Minimum Coverings of Pairs by Triples, with Application to 2-Radius Sequences

A new ordering, extending the notion of universal cycles of Chung {\em et al.} (1992), is proposed for the blocks of $k$-uniform set systems. Existence of minimum coverings of pairs by triples that possess such an ordering is established for all orders. Application to the construction of short 2-radius sequences is given, with some new 2-radius sequences found through computer search.

math.CO↗
arXiv2010

Query-Efficient Locally Decodable Codes of Subexponential Length

We develop the algebraic theory behind the constructions of Yekhanin (2008) and Efremenko (2009), in an attempt to understand the ``algebraic niceness'' phenomenon in $\mathbb{Z}_m$. We show that every integer $m = pq = 2^t -1$, where $p$, $q$ and $t$ are prime, possesses the same good algebraic property as $m=511$ that allows savings in query complexity. We identify 50 numbers of this form by computer search, which together with 511, are then applied to gain improvements on query complexity via Itoh and Suzuki's composition method. More precisely, we construct a $3^{\lceil r/2\rceil}$-query LDC for every positive integer $r<104$ and a $\left\lfloor (3/4)^{51}\cdot 2^{r}\right\rfloor$-query LDC for every integer $r\geq 104$, both of length $N_{r}$, improving the $2^r$ queries used by Efremenko (2009) and $3\cdot 2^{r-2}$ queries used by Itoh and Suzuki (2010). We also obtain new efficient private information retrieval (PIR) schemes from the new query-efficient LDCs.

cs.CC↗
arXiv2010

Linear Size Optimal q-ary Constant-Weight Codes and Constant-Composition Codes

An optimal constant-composition or constant-weight code of weight $w$ has linear size if and only if its distance $d$ is at least $2w-1$. When $d\geq 2w$, the determination of the exact size of such a constant-composition or constant-weight code is trivial, but the case of $d=2w-1$ has been solved previously only for binary and ternary constant-composition and constant-weight codes, and for some sporadic instances. This paper provides a construction for quasicyclic optimal constant-composition and constant-weight codes of weight $w$ and distance $2w-1$ based on a new generalization of difference triangle sets. As a result, the sizes of optimal constant-composition codes and optimal constant-weight codes of weight $w$ and distance $2w-1$ are determined for all such codes of sufficiently large lengths. This solves an open problem of Etzion. The sizes of optimal constant-composition codes of weight $w$ and distance $2w-1$ are also determined for all $w\leq 6$, except in two cases.

cs.IT↗