Algebraic Studies on Hadamard Matrices, Block Designs and Error-Correcting Codes

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(1) We have shown that there exists a regular even tournament for every positive integer v such that v is congruentto 3 modulo 8.(2) We have solved the existence problem for cyclic even tournaments. If v is congruent to 3 modulo 8, then a cyclic even tournament of order v exists if and only if 2 has a singly even order for every prime factor p of v. If v is congruent to 1 modulo 8, then a cyclic even tournament of order v exists if and only if 2 has an odd order for every prime factor p of v. In the second case an intimate relation with a binary cyclic code where the Hamming weight of every code word is a multiple of 4 exists.(3) We have determined the largest number which may be the order of the automorphism group G of a cyclic tournament of order v and the structure of G in such a case.(4) We have improved results of Alspach and Berggren concerning a tournament of order v whose automorphism group has the largest order. Namely we obtained necessary and sufficient conditions for v such that the group is a 3-group or (3, 5)-group.(5) We together with Raposa in Manila, Philippines have shown that the 3-intersection number pair is unique for a nearly triply regular symmetric design of RH-type.