Covering radius for sets of permutations

The Electronic Journal of Combinatorics, 2004

We present a class of permutations for which the number of distinctly ordered subsequences of each permutation approaches an almost optimal value as the length of the permutation grows to infinity.

Designs, Codes and Cryptography, 2012

Most papers on permutation codes have concentrated on the minimum Hamming distance of the code. An (n, d) permutation code (or permutation array) is simply a set of permutations on n elements in which the Hamming distance between any pair of distinct permutations (or codewords) is at least d. An (n, 2e + 1) or (n, 2e + 2) permutation code is able to correct up to e errors. These codes have a potential application to powerline communications. It is known that in an (n, 2e) permutation code the balls of radius e surrounding the codewords may all be pairwise disjoint, but usually some overlap. Thus an (n, 2e) permutation code is generally unable to correct e errors using nearest neighbour decoding. On the other hand, if the packing radius of the code is defined as the largest value of e for which the balls of radius e are all pairwise disjoint, a permutation code with packing radius e can be denoted by [n, e]. Such a permutation code can always correct e errors. In this paper it is shown that, in almost all cases considered, the number of codewords in the best [n, e] code found is substantially greater than the largest number of codewords in the best known (n, 2e + 1) code. Thus the packing radius more accurately specifies the requirement for an e-error-correcting permutation code than does the minimum Hamming distance. The techniques used include construction by automorphism group and several variations of clique search They are enhanced by two theoretical results which make the computations feasible.

1996

There are evident parallels between problems in design theory and coding theory. In design theory (see, eg, [3] and extensive list of references therein) we study k-element subsets and the covering relation is de ned as set-theoretic inclusion. In coding theory we deal with binary n-tuples and we say that one vector covers another if their Hamming distance (the number of coordinates in which they di er from each other) does not exceed some prescribed number.

Journal of Combinatorial Designs, 2009

Given l < k < n, a partial Steiner (n, k, l)-system P is a family of k-element subsets of an n-element set such that every l-element subset is contained in at most one member of P. The second author, followed by several others, verified a conjecture of P. Erdős and H. Hanani stating that, for every n, there exists a partial Steiner (n, k, l)-system of size (1−o(1)) n l / k l . While all known proofs involve the "semi-random method", in this short manuscript we use an algebraic construction (European J Combin 16(1) (1995), 35-40; Problems Control Inform Theory/Problemy Upravlen Teor Inform 12(1) (1983), 3-10) and complement it by an elementary counting argument to obtain a simple proof of the above fact. q

European Journal of Combinatorics, 2009

Let S(n) be the symmetric group on n points. A subset S of S(n) is intersecting if for any pair of permutations π, σ in S there is a point i ∈ {1, . . . , n} such that π(i) = σ(i). Deza and Frankl proved that if S ⊆ S(n) is intersecting then |S| ≤ (n − 1)!. Further, Cameron and Ku [4] show that the only sets that meet this bound are the cosets of a stabilizer of a point. In this paper we give a very different proof of this same result. * Research supported by NSERC.

IEEE Transactions on Information Theory, 1985

All known results on covering radius are presented, as well as some new results. There are a number of upper and lower bounds, including asymptotic results, a few exact determinations of covering radius, some extensive relations with other aspects of coding theory through the Reed-Muller codes, and new results on the least covering radius of any linear [II, k] code. There is also a recent result on the complexity of computing the covering radius.

Annals of Combinatorics, 2007

We study the length L k of the shortest permutation containing all patterns of length k. We establish the bounds e −2 k 2 < L k ≤ (2/3 + o(1))k 2. We also prove that as k → ∞, there are permutations of length (1/4 + o(1))k 2 containing almost all patterns of length k.

1986

We provide evidence for the following conjecture: a minimal covering of the binary Hamming space F~ + 2 by spheres of radius t + 1 has at most the same cardinality as a minimal covering of F~ by spheres of radius t. 1. Introduction For a vector x e F~, we denote by B(x ,t) the Hamming sphere of radius t centered at x, i.e. B(x,t) = {y e F~, d(x,y) ~ t}, where d(x ,y) is the Hamming distance between x and y. We say that C is a t-covering of F~ if C+B(O,t) =F~, where, for X , Y C F~, X + Y = {x + y , x eX, y eY}. Finally, let K(n,t) be the minimal cardinality of such a covering. If moreover C is a linear subspace of F~ (a linear code), we call it a linear t-covering, and denote by k[n,t] the minimal dimension of such a covering. We shall discuss the following conjectures.

Discrete Mathematics, 1991

In this paper we present a characterization of connected graphs of order (k + 1)n with k-covering number n, a characterization of trees in which the k-packing and k-covering numbers are the same, and we prove that the smallest number of subgraphs most k which cover the vertices of a block graph equals the k-packing number. of diameter at * This research was partially supported by the Heinrich Hertz Foundation and it was carried out while the author was visiting the Technical University of Aachen.

The electronic journal of combinatorics

Let S n be the set of all permutations on [n] := {1, 2, . . . , n}. We denote by κ n the smallest cardinality of a subset A of S n+1 that "covers" S n , in the sense that each π ∈ S n may be found as an orderisomorphic subsequence of some π ′ in A. What are general upper bounds on κ n ? If we randomly select ν n elements of S n+1 , when does the probability that they cover S n transition from 0 to 1? Can we provide a fine-magnification analysis that provides the "probability of coverage" when ν n is around the level given by the phase transition? In this paper we answer these questions and raise others.