Combinatorial design theory is a very active area of mathematical research, with many applications in communications and information theory, computer science, statistics, engineering, and life sciences. As one of the fundamental discrete structures, combinatorial designs are used in fields as diverse as error-correcting codes, statistical design of experiments, cryptography and information security, mobile and wireless communications, group testing algorithms in DNA screening, software and hardware testing, and interconnection networks. This monograph provides a tutorial on combinatorial designs, which gives an overview of the theory. Furthermore, the application of combinatorial designs to authentication and secrecy codes is described in depth. This close relationship of designs with cryptography and information security was first revealed in Shannon's seminal paper on secrecy systems. We bring together in one source foundational and current contributions concerning design-theoretic constructions and characterizations of authentication and secrecy codes.

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Combinatorial Designs for Authentication and Secrecy Codes

We propose a combination of two known computational methods for the construction of designs with prescribed groups of automorphisms: the Kramer–Mesner method and the method of tactical decompositions. This combined method is used to construct new unitals with parameters 2-(65, 5, 1). © 2011 Wiley Periodicals, Inc. J Combin Designs 19:290-303, 2011

The Kramer–Mesner method with tactical decompositions: some new unitals on 65 points

Group actions are reviewed as a tool for classifying combinatorial objects up to isomorphism. The objective is a general theory for constructing representatives of isomorphism types. Homomorphisms of group actions allow to reduce problem sizes step by step. In particular, classifying by stabilizer type, i.e. the automorphism group of the objects, is generalized to using only sufficiently large subgroups of stabilizers. So, less knowledge of the full subgroup lattice of the classifying group is needed. For single steps in the homomorphism decomposition, isomorphism problems are transformed into double coset problems in groups. New lower bounds are given for the number of long double cosets such that corresponding bounds for the number of objects with trivial automorphism group can be derived.

Constructing Objects up to Isomorphism, Simple 9-Designs with Small Parameters

We develop an algorithm framework for isomorph-free exhaustive generation of designs admitting a group of automorphisms from a prescribed collection of pairwise nonconjugate groups, where each prescribed group has a large index relative to its normalizer in the isomorphism-inducing group. We demonstrate the practicality of the framework by producing a complete classification of the Steiner triple systems of order $21$ admitting a nontrivial automorphism group. The number of such pairwise nonisomorphic designs is $62336617$, where $958$ of the designs are anti-Pasch. We also develop consistency checking methodology for gaining confidence in the correct operation of the algorithm implementation.

Isomorph-Free Exhaustive Generation of Designs with Prescribed Groups of Automorphisms

Designs usually provide examples for very hard isomorphism problems. So, instead of solving the general isomorphism problem here the problem is solved for many cases where the designs are constructed with a common prescribed group of automorphisms by Kramer and Mesner's approach [30]. The method is based on an analysis of group actions and improves on Burnside's table of marks approach [15]. In particular, no knowledge of the full subgroup lattice of the symmetric group on the point set is needed.

Solving isomorphism problems for t -designs

The first 5-(72, 6, 1) designs with automorphism group PSL(2, 71) were found by Mills [10]. We presently enumerate all 5-(72, 6, 1) designs with this automorphism group. There are in all 926299 non-isomorphic designs. We show that a necessary condition for semiregular5-(v, 6, 1) designs with automorphism group PSL(2, v 1) to exist is thatv=84, 228 (mod 360). In particular, there are exactly 3 non-isomorphic semiregular 5-(84, 6, 1) designs with automorphism group PSL(2, 83). There are at least 6450 non-isomorphic 5-(244, 6, 1) designs with automorphism group P∑L(2, 35).

Sergey Molodtsov

Papers

Steiner systems with automorphism groups PSL(2, 71), PSL(2, 83), and P∑L(2, 3 5 )