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 Optimal and Near-Optimal Multiply Constant-Weight Codes

Multiply constant-weight codes (MCWCs) have been recently studied to improve the reliability of certain physically unclonable function response. In this paper, we give combinatorial constructions for MCWCs which yield several new infinite families of optimal MCWCs. Furthermore, we demonstrate that the Johnson type upper bounds of MCWCs are asymptotically tight for fixed weights and distances. Finally, we provide bounds and constructions of two dimensional MCWCs

Jacobi polynomials and design theory II

In this paper, we introduce some new polynomials associated to linear codes over $\mathbb{F}_{q}$. In particular, we introduce the notion of split complete Jacobi polynomials attached to multiple sets of coordinate places of a linear code over $\mathbb{F}_{q}$, and give the MacWilliams type identity for it. We also give the notion of generalized $q$-colored $t$-designs. As an application of the generalized $q$-colored $t$-designs, we derive a formula that obtains the split complete Jacobi polynomials of a linear code over $\mathbb{F}_{q}$.Moreover, we define the concept of colored packing (resp. covering) designs. Finally, we give some coding theoretical applications of the colored designs for Type~III and Type~IV codes.Comment: 28 page

Generating Uniformly-Distributed Random Generalised 2-designs with Block Size 3

PhDGeneralised t-designs, defined by Cameron, describe a generalisation of many combinatorial objects including: Latin squares, 1-factorisations of K2n (the complete graph on 2n vertices), and classical t-designs. This new relationship raises the question of how their respective theory would fare in a more general setting. In 1991, Jacobson and Matthews published an algorithm for generating uniformly distributed random Latin squares and Cameron conjectures that this work extends to other generalised 2-designs with block size 3. In this thesis, we divide Cameron’s conjecture into three parts. Firstly, for constants RC, RS and CS, we study a generalisation of Latin squares, which are (r c) grids whose cells each contain RC symbols from the set f1;2; : : : ; sg such that each symbol occurs RS times in each column and CS times in each row. We give fundamental theory about these objects, including an enumeration for small parameter values. Further, we prove that Cameron’s conjecture is true for these designs, for all admissible parameter values, which provides the first method for generating them uniformly at random. Secondly, we look at a generalisation of 1-factorisations of the complete graph. For constants NN and NC, these graphs have n vertices, each incident with NN coloured edges, such that each colour appears at each vertex NC times. We successfully show how to generate these designs uniformly at random when NC 0 (mod 2) and NN NC. Finally, we observe the difficulties that arise when trying to apply Jacobson and Matthews’ theory to the classical triple systems. Cameron’s conjecture remains open for these designs, however, there is mounting evidence which suggests an affirmative result. A function reference for DesignMC, the bespoke software that was used during this research, is provided in an appendix