Overlarge sets of 2-(11, 5, 2) designs and related configurations

We consider the construction of several configurations, including: • overlarge sets of 2-(11,5,2) designs, that is, partitions of the set of all 5-subsets of a 12-set into 72 2-(11,5,2) designs; • an indecomposable doubly overlarge set of 2-(11,5,2) designs, that is, a partition of two copies of the set of all 5-subsets of a 12-set into 144 2-(11,5,2) designs, such that the 144 designs can be arranged into a 12 × 12 square with interesting row and column properties; • a partition of the Steiner system S(5,6,12) into 12 disjoint 2-(11,6,3) designs arising from the diagonal of the square; • bidistant permutation arrays and generalized Room squares arising from the doubly overlarge set, and their relation to some new strongly regular graphs.

Partitioning sets of triples into small planes

We study partitions of the set of all ((v)(3)) triples chosen from a v-set into pairwise disjoint planes with three points per line. Our partitions may contain copies of PG(2, 2) only (Fano partitions) or copies of AG(2, 3) only (affine partitions) or copies of some planes of each type (mixed partitions). We find necessary conditions for Fano or affine partitions to exist. Such partitions are already known in several cases: Fano partitions for v = 8 and affine partitions for v = 9 or 10. We construct such partitions for several sporadic orders, namely, Fano partitions for v = 14, 16, 22, 23, 28, and an affine partition for v = 18. Using these as starter partitions, we prove that Fano partitions exist for v = 7(n) + 1, 13(n) + 1, 27(n) + 1, and affine partitions for v = 8(n) + 1, 9(n) + 1, 17(n) + 1. In particular, both Fano and affine partitions exist for v = 3(6n) + 1. Using properties of 3-wise balanced designs, we extend these results to show that affine partitions also exist for v = 3(2n). Similarly, mixed partitions are shown to exist for v = 8(n), 9(n), 11(n) + 1.

The spectrum of minimal defining sets of some Steiner systems

For a design D, define spec(D) = {\M\ \ M is a minimal defining set of D} to be the spectrum of minimal defining sets of D. In this note we give bounds on the size of an element in spec(D) when D is a Steiner system. We also show that the spectrum of minimal defining sets of the Steiner triple system given by the points and lines of PG(3,2) equals {16,17,18,19,20,21,22}, and point out some open questions concerning the Steiner triple systems associated with PG(n, 2) in general. (C) 2002 Elsevier Science B.V. All rights reserved.

Skolem-type difference sets for cycle systems

Cyclic m-cycle systems of order v are constructed for all m greater than or equal to 3, and all v = 1(mod 2m). This result has been settled previously by several authors. In this paper, we provide a different solution, as a consequence of a more general result, which handles all cases using similar methods and which also allows us to prove necessary and sufficient conditions for the existence of a cyclic m-cycle system of K-v - F for all m greater than or equal to 3, and all v = 2(mod 2m).

Large sets of large sets of Steiner triple systems of order 9

We describe a direct method of partitioning the 840 Steiner triple systems of order 9 into 120 large sets. The method produces partitions in which all of the large sets are isomorphic and we apply the method to each of the two non-isomorphic large sets of STS(9).

Balanced critical sets in Latin squares

In this note we first introduce balanced critical sets and near balanced critical sets in Latin squares. Then we prove that there exist balanced critical sets in the back circulant Latin squares of order 3n for n even. Using this result we decompose the back circulant Latin squares of order 3n, n even, into three isotopic and disjoint balanced critical sets each of size 3n. We also find near balanced critical sets in the back circulant Latin squares of order 3n for n odd. Finally, we examine representatives of each main class of Latin squares of order up to six in order to determine which main classes contain balanced or near balanced critical sets.

A census of critical sets in the Latin squares of order at most six

A critical set in a Latin square of order n is a set of entries from the square which can be embedded in precisely one Latin square of order n, Such that if any element of the critical set. is deleted, the remaining set can be embedded, in more than one Latin square of order n.. In this paper we find all the critical sets of different sizes in the Latin squares of order at most six. We count the number of main and isotopy classes of these critical sets and classify critical sets from the main classes into various strengths. Some observations are made about the relationship between the numbers of classes, particularly in the 6 x 6 case. Finally some examples are given of each type of critical set.

An investigation of 2-critical sets in latin squares

In this paper we focus on the existence of 2-critical sets in the latin square corresponding to the elementary abelian 2-group of order 2(n). It has been shown by Stinson and van Rees that this latin square contains a 2-critical set of volume 4(n) - 3(n). We provide constructions for 2-critical sets containing 4(n) - 3(n) + 1 - (2(k-1) + 2(m-1) + 2(n-(k+m+1))) entries, where 1 less than or equal to k less than or equal to n and 1 less than or equal to m less than or equal to n - k. That is, we construct 2-critical sets for certain values less than 4(n) - 3(n) + 1 - 3 (.) 2([n /3]-1). The results raise the interesting question of whether, for the given latin square, it is possible to construct 2-critical sets of volume m, where 4(n) - 3(n) + 1 - 3 (.) 2([n/3]-1) < m < 4(n) - 3(n).

Mixed partitions of sets of triples into small planes

We continue our study of partitions of the set of all ((v)(3)) triples chosen from a v-set into pairwise disjoint planes with three points per line. We develop further necessary conditions for the existence of partitions of such sets into copies of PG(2, 2) and copies of AG(2, 3), and deal with the cases v = 13, 14, 15 and 17. These partitions, together with those already known for v = 12, 16 and 18, then become starters for recursive constructions of further infinite families of partitions. (C) 2004 Elsevier B.V. All rights reserved.

Searching oligo sets of human chromosome 12 using evolutionary strategies

DNA Microarray is a powerful tool to measure the level of a mixed population of nucleic acids at one time, which has great impact in many aspects of life sciences research. In order to distinguish nucleic acids with very similar composition by hybridization, it is necessary to design microarray probes with high specificities and sensitivities. Highly specific probes correspond to probes having unique DNA sequences; whereas highly sensitive probes correspond to those with melting temperature within a desired range and having no secondary structure. The selection of these probes from a set of functional DNA sequences (exons) constitutes a computationally expensive discrete non-linear search problem. We delegate the search task to a simple yet effective Evolution Strategy algorithm. The computational efficiency is also greatly improved by making use of an available bioinformatics tool.