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Cover-free families (CFFs) were considered from different subjects by numerous researchers. In this article, we mainly consider explicit constructions of (2; d)-cover-free families. We also determine the size of optimal 2-cover-free-families on 9, 10, and 11 points. Related separating hash families, which can be used to construct CFFs, are also discussed.

An LDðn; k; p; t; bÞ lotto design is a set of b k-sets (blocks) of an n-set such that any p-set intersects at least one k-set in t or more elements. Let Lðn; k; p; tÞ denote the minimum number of blocks in any LDðn; k; p; t; bÞ lotto design. We will list the known lower and upper bound theorems for lotto designs. Since many of these bounds are recursive, we will incorporate this information in a set of tables for lower and upper bounds for lotto designs with small parameters. We will also use back-track algorithms, greedy algorithms, and simulated annealing to improve the tables.

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Suppose that L is a latin square of order m and P ⊆ L is a partial latin square. If L is the only latin square of order m which contains P, and no proper subset of P has this property, then P is a critical set of L. The critical set spectrum problem is to determine, for a given m, the set of integers t for which there exists a latin square of order m with a critical set of size t. We outline a partial solution to the critical set spectrum problem for latin squares of order 2 n . The back circulant latin square of even order m has a well-known critical set of size m 2 /4, and this is the smallest known critical set for a latin square of order m. The abelian 2-group of order 2 n has a critical set of size 4 n − 3 n , and this is the largest known critical set for a latin square of order 2 n . We construct a set of latin squares with associated critical sets which are intermediate between the back circulant latin square of order 2 n and the abelian 2-group of order 2 n .

We enumerate a list of 594 inequivalent binary ð33; 16Þ doubly-even selforthogonal codes that have no all-zero coordinates along with their automorphism groups. It is proven that if a ð22; 8; 4Þ Balanced Incomplete Block Design were to exist then the 22 rows of its incident matrix will be contained in at least one of the 594 codes. Without using computers, we eliminate this possibility for 116 of these codes. # 2005 Wiley Periodicals, Inc. J Combin Designs 13: [363][364][365][366][367][368][369][370][371][372][373][374][375][376] 2005

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