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\u{\i}n 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.Comment: 23 page

Selected Papers in Combinatorics - a Volume Dedicated to R.G. Stanton

Professor Stanton has had a very illustrious career. His contributions to mathematics are varied and numerous. He has not only contributed to the mathematical literature as a prominent researcher but has fostered mathematics through his teaching and guidance of young people, his organizational skills and his publishing expertise. The following briefly addresses some of the areas where Ralph Stanton has made major contributions

Generalized packing designs

Generalized $t$-designs, which form a common generalization of objects such as $t$-designs, resolvable designs and orthogonal arrays, were defined by Cameron [P.J. Cameron, A generalisation of $t$-designs, \emph{Discrete Math.}\ {\bf 309} (2009), 4835--4842]. In this paper, we define a related class of combinatorial designs which simultaneously generalize packing designs and packing arrays. We describe the sometimes surprising connections which these generalized designs have with various known classes of combinatorial designs, including Howell designs, partial Latin squares and several classes of triple systems, and also concepts such as resolvability and block colouring of ordinary designs and packings, and orthogonal resolutions and colourings. Moreover, we derive bounds on the size of a generalized packing design and construct optimal generalized packings in certain cases. In particular, we provide methods for constructing maximum generalized packings with $t=2$ and block size $k=3$ or 4.Comment: 38 pages, 2 figures, 5 tables, 2 appendices. Presented at 23rd British Combinatorial Conference, July 201

Constructions of General Covering Designs

Given five positive integers $v, m,k,\lambda$ and $t$ where $v \geq k \geq t$ and $v \geq m \geq t,$ a $t$-$(v,k,m,\lambda)$ general covering design is a pair $(X,\mathcal{B})$ where $X$ is a set of $v$ elements (called points) and $\mathcal{B}$ a multiset of $k$-subsets of $X$ (called blocks) such that every $m$-subset of $X$ intersects (is covered by) at least $\lambda$ members of $\mathcal{B}$ in at least $t$ points. In this article we present new constructions for general covering designs and we generalize some others. By means of these constructions we will be able to obtain some new upper bounds on the minimum size of such designs.Comment: Section 3.2 revised and extended; plus some re-editing throughou

Decompositions of graphs and hypergraphs

This thesis contains various new results in the areas of design theory and edge decompositions of graphs and hypergraphs. Most notably, we give a new proof of the existence conjecture, dating back to the 19th century. For $r$-graphs $F$ and $G$, an $F$-decomposition of G is a collection of edge-disjoint copies of F in G covering all edges of $G$. In a recent breakthrough, Keevash proved that every sufficiently large quasirandom $r$-graph G has a $K$$_f$$^{(r)}$ -decomposition (subject to necessary divisibility conditions), thus proving the existence conjecture. We strengthen Keevash's result in two major directions: Firstly, our main result applies to decompositions into any $r$-graph $F$, which generalises a fundamental theorem of Wilson to hypergraphs. Secondly, our proof framework applies beyond quasirandomness, enabling us e.g. to deduce a minimum degree version. For graphs, we investigate the minimum degree setting further. In particular, we determine the decomposition threshold' of every bipartite graph, and show that the threshold of cliques is equal to its fractional analogue. We also present theorems concerning optimal path and cycle decompositions of quasirandom graphs. This thesis is based on joint work with Daniela Kuhn and Deryk Osthus, Allan Lo and Richard Montgomery