On Perfect Codes in the Johnson Graph

In this paper we consider the existence of nontrivial perfect codes in the Johnson graph J(n,w). We present combinatorial and number theory techniques to provide necessary conditions for existence of such codes and reduce the range of parameters in which 1-perfect and 2-perfect codes may exist.Comment: Submitted for ACCT 201

Properties of Codes in the Johnson Scheme

Codes which attain the sphere packing bound are called perfect codes. The most important metrics in coding theory on which perfect codes are defined are the Hamming metric and the Johnson metric. While for the Hamming metric all perfect codes over finite fields are known, in the Johnson metric it was conjectured by Delsarte in 1970's that there are no nontrivial perfect codes. The general nonexistence proof still remains the open problem. In this work we examine constant weight codes as well as doubly constant weight codes, and reduce the range of parameters in which perfect codes may exist in both cases. We start with the constant weight codes. We introduce an improvement of Roos' bound for one-perfect codes, and present some new divisibility conditions, which are based on the connection between perfect codes in Johnson graph J(n,w) and block designs. Next, we consider binomial moments for perfect codes. We show which parameters can be excluded for one-perfect codes. We examine two-perfect codes in J(2w,w) and present necessary conditions for existence of such codes. We prove that there are no two-perfect codes in J(2w,w) with length less then 2.5*10^{15}. Next we examine perfect doubly constant weight codes. We present a family of parameters for codes whose size of sphere divides the size of whole space. We then prove a bound on length of such codes, similarly to Roos' bound for perfect codes in Johnson graph. Finally we describe Steiner systems and doubly Steiner systems, which are strongly connected with the constant weight and doubly constant weight codes respectively. We provide an anticode-based proof of a bound on length of Steiner system, prove that doubly Steiner system is a diameter perfect code and present a bound on length of doubly Steiner system.Comment: This is an M.Sc.thesis submitted in February 2007 by Natalia Silberstein and supervised by Prof. Tuvi Etzio

Improved List-Decodability of Reed--Solomon Codes via Tree Packings

This paper shows that there exist Reed--Solomon (RS) codes, over large finite fields, that are combinatorially list-decodable well beyond the Johnson radius, in fact almost achieving list-decoding capacity. In particular, we show that for any $\epsilon\in (0,1]$ there exist RS codes with rate $\Omega(\frac{\epsilon}{\log(1/\epsilon)+1})$ that are list-decodable from radius of $1-\epsilon$. We generalize this result to obtain a similar result on list-recoverability of RS codes. Along the way we use our techniques to give a new proof of a result of Blackburn on optimal linear perfect hash matrices, and strengthen it to obtain a construction of strongly perfect hash matrices. To derive the results in this paper we show a surprising connection of the above problems to graph theory, and in particular to the tree packing theorem of Nash-Williams and Tutte. En route to our results on RS codes, we prove a generalization of the tree packing theorem to hypergraphs (and we conjecture that an even stronger generalization holds). We hope that this generalization to hypergraphs will be of independent interest.Comment: 47 pages, submitte

Separating hash families: A Johnson-type bound and new constructions

Separating hash families are useful combinatorial structures which are generalizations of many well-studied objects in combinatorics, cryptography and coding theory. In this paper, using tools from graph theory and additive number theory, we solve several open problems and conjectures concerning bounds and constructions for separating hash families. Firstly, we discover that the cardinality of a separating hash family satisfies a Johnson-type inequality. As a result, we obtain a new upper bound, which is superior to all previous ones. Secondly, we present a construction for an infinite class of perfect hash families. It is based on the Hamming graphs in coding theory and generalizes many constructions that appeared before. It provides an affirmative answer to both Bazrafshan-Trung's open problem on separating hash families and Alon-Stav's conjecture on parent-identifying codes. Thirdly, let $p_t(N,q)$ denote the maximal cardinality of a $t$-perfect hash family of length $N$ over an alphabet of size $q$. Walker II and Colbourn conjectured that $p_3(3,q)=o(q^2)$. We verify this conjecture by proving $q^{2-o(1)}<p_3(3,q)=o(q^2)$. Our proof can be viewed as an application of Ruzsa-Szemer{\'e}di's (6,3)-theorem. We also prove $q^{2-o(1)}<p_4(4,q)=o(q^2)$. Two new notions in graph theory and additive number theory, namely rainbow cycles and $R$-sum-free sets, are introduced to prove this result. These two bounds support a question of Blackburn, Etzion, Stinson and Zaverucha. Finally, we establish a bridge between perfect hash families and hypergraph Tur{\'a}n problems. This connection has not been noticed before. As a consequence, many new results and problems arise.Comment: 20 pages, accepted in SIAM Journal on Discrete Mathematic

Perfect codes in Doob graphs

We study $1$-perfect codes in Doob graphs $D(m,n)$. We show that such codes that are linear over $GR(4^2)$ exist if and only if $n=(4^{g+d}-1)/3$ and $m=(4^{g+2d}-4^{g+d})/6$ for some integers $g \ge 0$ and $d>0$. We also prove necessary conditions on $(m,n)$ for $1$-perfect codes that are linear over $Z_4$ (we call such codes additive) to exist in $D(m,n)$ graphs; for some of these parameters, we show the existence of codes. For every $m$ and $n$ satisfying $2m+n=(4^t-1)/3$ and $m \le (4^t-5\cdot 2^{t-1}+1)/9$, we prove the existence of $1$-perfect codes in $D(m,n)$, without the restriction to admit some group structure. Keywords: perfect codes, Doob graphs, distance regular graphs.Comment: 11p

The existence of perfect codes in Doob graphs

We solve the problem of existence of perfect codes in the Doob graph. It is shown that 1-perfect codes in the Doob graph D(m,n) exist if and only if 6m+3n+1 is a power of 2; that is, if the size of a 1-ball divides the number of vertices

On weight distributions of perfect colorings and completely regular codes

A vertex coloring of a graph is called "perfect" if for any two colors $a$ and $b$, the number of the color-$b$ neighbors of a color-$a$ vertex $x$ does not depend on the choice of $x$, that is, depends only on $a$ and $b$ (the corresponding partition of the vertex set is known as "equitable"). A set of vertices is called "completely regular" if the coloring according to the distance from this set is perfect. By the "weight distribution" of some coloring with respect to some set we mean the information about the number of vertices of every color at every distance from the set. We study the weight distribution of a perfect coloring (equitable partition) of a graph with respect to a completely regular set (in particular, with respect to a vertex if the graph is distance-regular). We show how to compute this distribution by the knowledge of the color composition over the set. For some partial cases of completely regular sets, we derive explicit formulas of weight distributions. Since any (other) completely regular set itself generates a perfect coloring, this gives universal formulas for calculating the weight distribution of any completely regular set from its parameters. In the case of Hamming graphs, we prove a very simple formula for the weight enumerator of an arbitrary perfect coloring. Codewords: completely regular code; equitable partition; partition design; perfect coloring; perfect structure; regular partition; weight distribution; weight enumerator.Comment: 17pp; partially presented at "Optimal Codes and Related Topics" OC2009, Varna (Bulgaria). V.2: the title was changed (old: "On weight distributions of perfect structures"), Sect.5 "Weight enumerators ..." was adde

Perfect Codes in the Discrete Simplex

We study the problem of existence of (nontrivial) perfect codes in the discrete $n$-simplex $\Delta_{\ell}^n := \left\{ \begin{pmatrix} x_0, \ldots, x_n \end{pmatrix} : x_i \in \mathbb{Z}_{+}, \sum_i x_i = \ell \right\}$ under $\ell_1$ metric. The problem is motivated by the so-called multiset codes, which have recently been introduced by the authors as appropriate constructs for error correction in the permutation channels. It is shown that $e$-perfect codes in the $1$-simplex $\Delta_{\ell}^1$ exist for any $\ell \geq 2e + 1$, the $2$-simplex $\Delta_{\ell}^2$ admits an $e$-perfect code if and only if $\ell = 3e + 1$, while there are no perfect codes in higher-dimensional simplices. In other words, perfect multiset codes exist only over binary and ternary alphabets.Comment: 15 pages (single-column), 5 figures. Minor revisions made. Accepted for publication in Designs, Codes and Cryptograph

MMS-type problems for Johnson scheme

In the current work we consider the minimization problems for the number of nonzero or negative values of vectors from the first and second eigenspaces of the Johnson scheme respectively. The topic is a meeting point for generalizations of the Manikam-Mikl\'{o}s-Singhi conjecture proven by Blinovski and the minimum support problem for the eigenspaces of the Johnson graph, asymptotically solved by authors in a recent paper.Comment: 9 page

The extended 1-perfect trades in small hypercubes

An extended $1$-perfect trade is a pair $(T_0,T_1)$ of two disjoint binary distance-$4$ even-weight codes such that the set of words at distance $1$ from $T_0$ coincides with the set of words at distance $1$ from $T_1$. Such trade is called primary if any pair of proper subsets of $T_0$ and $T_1$ is not a trade. Using a computer-aided approach, we classify nonequivalent primary extended $1$-perfect trades of length $10$, constant-weight extended $1$-perfect trades of length $12$, and Steiner trades derived from them. In particular, all Steiner trades with parameters $(5,6,12)$ are classified.Comment: 18pp. v.3: revised; bitrades are called trades in v.3; more reference