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Minimum Sum and Difference Covers of Abelian Groups
, 2004
"... A subset S of a finite Abelian group G is said to be a sum cover of G if every element of G can be expressed as the sum of two not necessarily distinct elements in S, a strict sum cover of G if every element of G can be expressed as the sum of two distinct elements in S, and a difference cover of G ..."
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A subset S of a finite Abelian group G is said to be a sum cover of G if every element of G can be expressed as the sum of two not necessarily distinct elements in S, a strict sum cover of G if every element of G can be expressed as the sum of two distinct elements in S, and a difference cover of G if every element of G can be expressed as the difference of two elements in S. For each type of cover, we determine for small k the largest Abelian group for which a kelement cover exists. For this purpose we compute a minimum sum cover, a minimum strict sum cover, and a minimum difference cover for Abelian groups of order up to 85, 90, and 127, respectively, by a backtrack search with isomorph rejection.
The Near Resolvable 2(13, 4, 3) Designs and ThirteenPlayer Whist Tournaments
 Codes Cryptogr
, 2004
"... A vplayer whist tournament is a schedule of games, where in each round the v players are partitioned into games of four players each with at most one player left over. In each game two of the players play as partners against the other two. All pairs of players must play in the same game exactly thr ..."
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Cited by 4 (1 self)
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A vplayer whist tournament is a schedule of games, where in each round the v players are partitioned into games of four players each with at most one player left over. In each game two of the players play as partners against the other two. All pairs of players must play in the same game exactly three times during the tournament; of those three times, they are to play as partners exactly once. Whist tournaments for v players are known to exist for all v &equiv; 0, 1 (mod 4). The special cases of directed whist tournaments and triplewhist tournaments are known to exist for all sufficiently large v, but for small v several open cases remain. In this paper we introduce a correspondence between near resolvable 2(v, k, &lambda;) designs and a particular class of codes. The near resolvable 2(13, 4, 3) designs are classified by classifying the corresponding codes with an orderly algorithm. Finally, the thirteenplayer whist tournaments are enumerated starting from the near resolvable 2(13, 4, 3) designs.
Discrete Mathematics for Combinatorial Chemistry
, 1998
"... The aim is a description of discrete mathematics used in a project devoted to the implementation of a software package for the simulation of combinatorial chemistry. ..."
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Cited by 2 (1 self)
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The aim is a description of discrete mathematics used in a project devoted to the implementation of a software package for the simulation of combinatorial chemistry.
Algorithmic Approach to Nonsymmetric 3class Association Schemes
"... Summary. There are 24 feasible parameter sets for a primitive nonsymmetric association schemes with 3 classes and at most 100 vertices. Using computer search, we prove nonexistence for three feasible parameter sets. Eleven cases are still open. In the imprimitive case, we survey the known results ..."
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Summary. There are 24 feasible parameter sets for a primitive nonsymmetric association schemes with 3 classes and at most 100 vertices. Using computer search, we prove nonexistence for three feasible parameter sets. Eleven cases are still open. In the imprimitive case, we survey the known results including some constructions of infinite families of schemes. In the smallest case that has been open up to now, we use computer search to find new schemes. These schemes are equivalent to “skew ” Bushtype Hadamard matrices of order 36. We also consider directed graphs that satisfy only some of the conditions required for a nonsymmetric association scheme with 3 classes. 1
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"... Abstract. In the paper “A new class of codes for Boolean masking of cryptographic computations, ” Carlet, Gaborit, Kim, and Solé defined a new class of rate onehalf binary codes called complementary information set (or CIS) codes. The authors then classified all CIS codes of length less than or equ ..."
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Abstract. In the paper “A new class of codes for Boolean masking of cryptographic computations, ” Carlet, Gaborit, Kim, and Solé defined a new class of rate onehalf binary codes called complementary information set (or CIS) codes. The authors then classified all CIS codes of length less than or equal to 12. CIS codes have relations to classical Coding Theory as they are a generalization of selfdual codes. As stated in the paper, CIS codes also have important practical applications as they may improve the cost of masking cryptographic algorithms against side channel attacks. In this paper, we give a complete classification result for length 14 CIS codes using an equivalence relation on GL(n, F2). We also give a new classification for all binary [16, 8, 3] and [16, 8, 4] codes. We then complete the classification for length 16 CIS codes and give additional classifications for optimal CIS codes of lengths 20 and 26. 1. Motivations A generalization of selfdual codes was recently proposed by Carlet, Gaborit, Kim, and Solé in [5]. In the paper, a new class of codes, called complementary
Rank of Adjacency Matrices of Directed (Strongly) Regular Graphs
"... For a positive integer r we consider the set Br of all values of k n for which there exists an n ×n matrix with entries 0 and 1 such that each row and each column has exactly k 1’s and the matrix has rank r. We prove that the set Br is finite, for every r. If there exists a kregular directed graph ..."
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For a positive integer r we consider the set Br of all values of k n for which there exists an n ×n matrix with entries 0 and 1 such that each row and each column has exactly k 1’s and the matrix has rank r. We prove that the set Br is finite, for every r. If there exists a kregular directed graph on n vertices such that its adjacency matrix has rank r then k n ∈ Br. We use this to exclude existence of directed strongly regular graphs for infinitely many feasible parameter sets. The investigation of {0, 1} matrices of low rank in this paper is motivated by the study of directed strongly regular graphs, which were introduced by A. Duval [2], see section 3. Fiedler, Klin and Muzychuk [4] excluded the existence of a 6regular directed strongly regular graph on 16 vertices for
Lehrstuhl II für Mathematik (Informatik)
, 2003
"... The generation of molecular graphs by computer programs has undergone some changes. The development is reported with focus on various mathematical methods that are created and employed in this process. No attempt has been made to explicitely state and prove the theorems but this overview contains hi ..."
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The generation of molecular graphs by computer programs has undergone some changes. The development is reported with focus on various mathematical methods that are created and employed in this process. No attempt has been made to explicitely state and prove the theorems but this overview contains hints to the relevant literature. In particular, a new generator MOLGEN 4.0 is described that aims at highest flexibility in using constraints during the generation process and, thus, meeting the needs of practical applications.