On the volumes and affine types of trades

A $[t]$-trade is a pair $T=(T_+, T_-)$ of disjoint collections of subsets (blocks) of a $v$-set $V$ such that for every $0\le i\le t$, any $i$-subset of $V$ is included in the same number of blocks of $T_+$ and of $T_-$. It follows that $|T_+| = |T_-|$ and this common value is called the volume of $T$. If we restrict all the blocks to have the same size, we obtain the classical $t$-trades as a special case of $[t]$-trades. It is known that the minimum volume of a nonempty $[t]$-trade is $2^t$. Simple $[t]$-trades (i.e., those with no repeated blocks) correspond to a Boolean function of degree at most $v-t-1$. From the characterization of Kasami--Tokura of such functions with small number of ones, it is known that any simple $[t]$-trade of volume at most $2\cdot2^t$ belongs to one of two affine types, called Type\,(A) and Type\,(B) where Type\,(A) $[t]$-trades are known to exist. By considering the affine rank, we prove that $[t]$-trades of Type\,(B) do not exist. Further, we derive the spectrum of volumes of simple trades up to $2.5\cdot 2^t$, extending the known result for volumes less than $2\cdot 2^t$. We also give a characterization of "small" $[t]$-trades for $t=1,2$. Finally, an algorithm to produce $[t]$-trades for specified $t$, $v$ is given. The result of the implementation of the algorithm for $t\le4$, $v\le7$ is reported.Comment: 30 pages, final version, to appear in Electron. J. Combi

Algorithms for classification of combinatorial objects

A recurrently occurring problem in combinatorics is the need to completely characterize a finite set of finite objects implicitly defined by a set of constraints. For example, one could ask for a list of all possible ways to schedule a football tournament for twelve teams: every team is to play against every other team during an eleven-round tournament, such that every team plays exactly one game in every round. Such a characterization is called a classification for the objects of interest. Classification is typically conducted up to a notion of structural equivalence (isomorphism) between the objects. For example, one can view two tournament schedules as having the same structure if one can be obtained from the other by renaming the teams and reordering the rounds. This thesis examines algorithms for classification of combinatorial objects up to isomorphism. The thesis consists of five articles – each devoted to a specific family of objects – together with a summary surveying related research and emphasizing the underlying common concepts and techniques, such as backtrack search, isomorphism (viewed through group actions), symmetry, isomorph rejection, and computing isomorphism. From an algorithmic viewpoint the focus of the thesis is practical, with interest on algorithms that perform well in practice and yield new classification results; theoretical properties such as the asymptotic resource usage of the algorithms are not considered. The main result of this thesis is a classification of the Steiner triple systems of order 19. The other results obtained include the nonexistence of a resolvable 2-(15, 5, 4) design, a classification of the one-factorizations of k-regular graphs of order 12 for k ≤ 6 and k = 10, 11, a classification of the near-resolutions of 2-(13, 4, 3) designs together with the associated thirteen-player whist tournaments, and a classification of the Steiner triple systems of order 21 with a nontrivial automorphism group.reviewe