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On the structure and classification of SOMAs: generalizations of mutually orthogonal Latin squares
 Electronic Journal of Combinatorics
, 1999
"... Let k 0 and n 2 be integers. A SOMA, or more specifically a SOMA(k;n), is an n \Theta n array A, whose entries are ksubsets of a knset\Omega\Gamma such that each element of\Omega occurs exactly once in each row and exactly once in each column of A, and no 2subset of\Omega is contained in more ..."
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Let k 0 and n 2 be integers. A SOMA, or more specifically a SOMA(k;n), is an n \Theta n array A, whose entries are ksubsets of a knset\Omega\Gamma such that each element of\Omega occurs exactly once in each row and exactly once in each column of A, and no 2subset of\Omega is contained in more than one entry of A. A SOMA(k;n) can be constructed by superposing k mutually orthogonal Latin squares of order n with pairwise disjoint symbolsets, and so a SOMA(k;n) can be seen as a generalization of k mutually orthogonal Latin squares of order n. In this paper we first study the structure of SOMAs, concentrating on how SOMAs can decompose. We then report on the use of computational group theory and graph theory in the discovery and classification of SOMAs. In particular, we discover and classify SOMA(3; 10)s with certain properties, and discover two SOMA(4; 14)s (SOMAs with these parameters were previously unknown to exist). Some of the newly discovered SOMA(3; 10)s come from superpos...
Codes for Interactive Authentication

, 1998
"... An authentication protocol is a procedure by which an informant tries to convey n bits of information, which we call an input message, to a recipient. An intruder, I, controls the network over which the informant and the recipient talk and may change any message before it reaches its destination ..."
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Cited by 12 (1 self)
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An authentication protocol is a procedure by which an informant tries to convey n bits of information, which we call an input message, to a recipient. An intruder, I, controls the network over which the informant and the recipient talk and may change any message before it reaches its destination. a If the protocol ha security p, then the the recipient must detect this a cheating with probability at leat I  p. This paper
Kirkman School Project Designs
"... A Kirkman school project design on v elements consists of the maximum admissible number of disjoint parallel classes, each containing blocks of sizes three except possibly one of size two or four. Cern'y, Hor'ak, and Wallis completely settled existence when v j 0; 2 (mod 3) and made some progress a ..."
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Cited by 2 (0 self)
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A Kirkman school project design on v elements consists of the maximum admissible number of disjoint parallel classes, each containing blocks of sizes three except possibly one of size two or four. Cern'y, Hor'ak, and Wallis completely settled existence when v j 0; 2 (mod 3) and made some progress and advanced a conjecture when v j 1 (mod 3). In this paper, a complete solution for the existence of such designs when v j 4 (mod 6) is given, and a nearly complete solution when v j 1 (mod 6) is also given. 1 Introduction A group divisible design (GDD) is a triple (X; G; B) which satisfies the following properties: (1) G is a partition of a set X (of points) into subsets called groups, (2) B is a set of subsets of X (blocks) such that a group and a block contain at most one common point, (3) every pair of points from distinct groups occurs in a unique block. The grouptype (type) of the GDD is the multiset fjGj : G 2 Gg. We usually use an "exponential" notation to describe grouptype: gr...
New combinatorial designs and their applications to authentication codes and secret sharing schemes. Discrete Mathematics 279
, 2004
"... Abstract. This paper introduces three new types of combinatorial designs, which we call external difference families (EDF), external BIBDs (EBIBD) and splitting BIBDs. An EDF is a special type of EBIBD, so existence of an EDF implies existence of an EBIBD. We construct optimal splitting Acodes by u ..."
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Abstract. This paper introduces three new types of combinatorial designs, which we call external difference families (EDF), external BIBDs (EBIBD) and splitting BIBDs. An EDF is a special type of EBIBD, so existence of an EDF implies existence of an EBIBD. We construct optimal splitting Acodes by using EDF. Then we give a new bound on the number of shares required in robust secret sharing schemes (i.e., schemes secure against cheaters). EDF can be used to construct robust secret sharing schemes that are optimal with respect to the new bound. We also prove a weak converse, showing that if there exists an optimal secret sharing scheme, then there exists an EBIBD. Finally, we derive a Fishertype inequality for splitting BIBDs. We also prove a weak equivalence between splitting BIBDs and splitting Acodes. Further, it is shown that an EDF implies a splitting BIBD.
New Combinatorial Designs and their Applications to Authentication Codes and Secret Sharing Schemes
, 2003
"... This paper introduces three new types of combinatorial designs, which we call external di#erence families (EDF), external BIBDs (EBIBD) and splitting BIBDs. An EDF is a special type of EBIBD, so existence of an EDF implies existence of an EBIBD. We construct optimal splitting Acodes by using EDF. T ..."
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This paper introduces three new types of combinatorial designs, which we call external di#erence families (EDF), external BIBDs (EBIBD) and splitting BIBDs. An EDF is a special type of EBIBD, so existence of an EDF implies existence of an EBIBD. We construct optimal splitting Acodes by using EDF. Then we give a new bound on the number of shares required in robust secret sharing schemes (i.e., schemes secure against cheaters). EDF can be used to construct robust secret sharing schemes that are optimal with respect to the new bound. We also prove a weak converse, showing that if there exists an optimal secret sharing scheme, then there exists an EBIBD. Finally, we derive a Fishertype inequality for splitting BIBDs. We also prove a weak equivalence between splitting BIBDs and splitting Acodes. Further, it is shown that an EDF implies a splitting BIBD.