New Constant-Weight Codes from Propagation Rules
This paper proposes some simple propagation rules which give rise to new binary constant-weight codes.
cs.IT↗
arXiv subjects
Yeow Meng Chee
Publications and source records attributed to Yeow Meng Chee.
Optimal Partitioned Cyclic Difference Packings for Frequency Hopping and Code Synchronization
Optimal partitioned cyclic difference packings (PCDPs) are shown to give rise to optimal frequency-hopping sequences and optimal comma-free codes. New constructions for PCDPs, based on almost difference sets and cyclic difference matrices, are given. These produce new infinite families of optimal PCDPs (and hence optimal frequency-hopping sequences and optimal comma-free codes). The existence problem for optimal PCDPs in ${\mathbb Z}_{3m}$, with $m$ base blocks of size three, is also solved for all $m\not\equiv 8,16\pmod{24}$.
cs.IT↗
Spectrum of Sizes for Perfect Deletion-Correcting Codes
One peculiarity with deletion-correcting codes is that perfect $t$-deletion-correcting codes of the same length over the same alphabet can have different numbers of codewords, because the balls of radius $t$ with respect to the Levenshte\uın distance may be of different sizes. There is interest, therefore, in determining all possible sizes of a perfect $t$-deletion-correcting code, given the length $n$ and the alphabet size~$q$. In this paper, we determine completely the spectrum of possible sizes for perfect $q$-ary 1-deletion-correcting codes of length three for all $q$, and perfect $q$-ary 2-deletion-correcting codes of length four for almost all $q$, leaving only a small finite number of cases in doubt.
cs.IT↗
Limit on the Addressability of Fault-Tolerant Nanowire Decoders
Although prone to fabrication error, the nanowire crossbar is a promising candidate component for next generation nanometer-scale circuits. In the nanowire crossbar architecture, nanowires are addressed by controlling voltages on the mesowires. For area efficiency, we are interested in the maximum number of nanowires $N(m,e)$ that can be addressed by $m$ mesowires, in the face of up to $e$ fabrication errors. Asymptotically tight bounds on $N(m,e)$ are established in this paper. In particular, it is shown that $N(m,e) = Θ(2^m / m^{e+1/2})$. Interesting observations are made on the equivalence between this problem and the problem of constructing optimal EC/AUED codes, superimposed distance codes, pooling designs, and diffbounded set systems. Results in this paper also improve upon those in the EC/AUEC codes literature.
cs.AR↗
Strongly Multiplicative and 3-Multiplicative Linear Secret Sharing Schemes
Strongly multiplicative linear secret sharing schemes (LSSS) have been a powerful tool for constructing secure multiparty computation protocols. However, it remains open whether or not there exist efficient constructions of strongly multiplicative LSSS from general LSSS. In this paper, we propose the new concept of a 3-multiplicative LSSS, and establish its relationship with strongly multiplicative LSSS. More precisely, we show that any 3-multiplicative LSSS is a strongly multiplicative LSSS, but the converse is not true; and that any strongly multiplicative LSSS can be efficiently converted into a 3-multiplicative LSSS. Furthermore, we apply 3-multiplicative LSSS to the computation of unbounded fan-in multiplication, which reduces its round complexity to four (from five of the previous protocol based on strongly multiplicative LSSS). We also give two constructions of 3-multiplicative LSSS from Reed-Muller codes and algebraic geometric codes. We believe that the construction and verification of 3-multiplicative LSSS are easier than those of strongly multiplicative LSSS. This presents a step forward in settling the open problem of efficient constructions of strongly multiplicative LSSS from general LSSS.
cs.CR↗
Group Divisible Codes and Their Application in the Construction of Optimal Constant-Composition Codes of Weight Three
The concept of group divisible codes, a generalization of group divisible designs with constant block size, is introduced in this paper. This new class of codes is shown to be useful in recursive constructions for constant-weight and constant-composition codes. Large classes of group divisible codes are constructed which enabled the determination of the sizes of optimal constant-composition codes of weight three (and specified distance), leaving only four cases undetermined. Previously, the sizes of constant-composition codes of weight three were known only for those of sufficiently large length.
cs.IT↗
The fine intersection problem for Steiner triple systems
The intersection of two Steiner triple systems (X,A) and (X,B) is the set A intersect B. The fine intersection problem for Steiner triple systems is to determine for each v, the set I(v), consisting of all possible pairs (m,n) such that there exist two Steiner triple systems of order v whose intersection has n blocks over m points. We show that for v = 1 or 3 (mod 6), |I(v)| = Omega(v^3), where previous results only imply that |I(v)| = Omega(v^2).
math.CO↗
Improved Lower Bounds for Constant GC-Content DNA Codes
The design of large libraries of oligonucleotides having constant GC-content and satisfying Hamming distance constraints between oligonucleotides and their Watson-Crick complements is important in reducing hybridization errors in DNA computing, DNA microarray technologies, and molecular bar coding. Various techniques have been studied for the construction of such oligonucleotide libraries, ranging from algorithmic constructions via stochastic local search to theoretical constructions via coding theory. We introduce a new stochastic local search method which yields improvements up to more than one third of the benchmark lower bounds of Gaborit and King (2005) for n-mer oligonucleotide libraries when n <= 14. We also found several optimal libraries by computing maximum cliques on certain graphs.
cs.IT↗
The Sizes of Optimal q-Ary Codes of Weight Three and Distance Four: A Complete Solution
This correspondence introduces two new constructive techniques to complete the determination of the sizes of optimal q-ary codes of constant weight three and distance four.
cs.IT↗
An Enumeration of Graphical Designs
Let $Ψ(t,k)$ denote the set of pairs $(v,λ)$ for which there exists a graphical $t$-$(v,k,λ)$ design. Most results on graphical designs have gone to show the finiteness of $Ψ(t,k)$ when $t$ and $k$ satisfy certain conditions. The exact determination of $Ψ(t,k)$ for specified $t$ and $k$ is a hard problem and only $Ψ(2,3)$, $Ψ(2,4)$, $Ψ(3,4)$, $Ψ(4,5)$, and $Ψ(5,6)$ have been determined. In this paper, we determine completely the sets $Ψ(2,5)$ and $Ψ(3,5)$. As a result, we find more than 270000 inequivalent graphical designs, and more than 8000 new parameter sets for which there exists a graphical design. Prior to this, graphical designs are known for only 574 parameter sets.
math.CO↗
A New Lower Bound for A(17,6,6)
We construct a record-breaking binary code of length 17, minimal distance 6, constant weight 6, and containing 113 codewords.
cs.IT↗
On Extremal k-Graphs Without Repeated Copies of 2-Intersecting Edges
The problem of determining extremal hypergraphs containing at most r isomorphic copies of some element of a given hypergraph family was first studied by Boros et al. in 2001. There are not many hypergraph families for which exact results are known concerning the size of the corresponding extremal hypergraphs, except for those equivalent to the classical Turan numbers. In this paper, we determine the size of extremal k-uniform hypergraphs containing at most one pair of 2-intersecting edges for k in {3,4}. We give a complete solution when k=3 and an almost complete solution (with eleven exceptions) when k=4.
math.CO↗
Optimal Memoryless Encoding for Low Power Off-Chip Data Buses
Off-chip buses account for a significant portion of the total system power consumed in embedded systems. Bus encoding schemes have been proposed to minimize power dissipation, but none has been demonstrated to be optimal with respect to any measure. In this paper, we give the first provably optimal and explicit (polynomial-time constructible) families of memoryless codes for minimizing bit transitions in off-chip buses. Our results imply that having access to a clock does not make a memoryless encoding scheme that minimizes bit transitions more powerful.
cs.AR↗
Constructions for Difference Triangle Sets
Difference triangle sets are useful in many practical problems of information transmission. This correspondence studies combinatorial and computational constructions for difference triangle sets having small scopes. Our algorithms have been used to produce difference triangle sets whose scopes are the best currently known.
cs.IT↗
The PBD-Closure of Constant-Composition Codes
We show an interesting PBD-closure result for the set of lengths of constant-composition codes whose distance and size meet certain conditions. A consequence of this PBD-closure result is that the size of optimal constant-composition codes can be determined for infinite families of parameter sets from just a single example of an optimal code. As an application, the size of several infinite families of optimal constant-composition codes are derived. In particular, the problem of determining the size of optimal constant-composition codes having distance four and weight three is solved for all lengths sufficiently large. This problem was previously unresolved for odd lengths, except for lengths seven and eleven.
cs.IT↗
Constructions of q-Ary Constant-Weight Codes
This paper introduces a new combinatorial construction for q-ary constant-weight codes which yields several families of optimal codes and asymptotically optimal codes. The construction reveals intimate connection between q-ary constant-weight codes and sets of pairwise disjoint combinatorial designs of various types.
cs.IT↗