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

(1)证明了每个正整数v都存在正则偶竞赛图，使得v与3模8同余。(2)解决了循环偶竞赛图的存在性问题。如果v与3模8同余，则存在v阶的循环偶数竞赛赛当且仅当2对v的每个素数因子p有单偶阶，如果v与1模8同余，则存在v阶循环偶数竞赛当且仅当2对v的每个素数因子p有奇数阶。在第二种情况下，存在与二元循环码的密切关系，其中每个码字的汉明重量是4的倍数。(3)我们确定了v阶循环竞赛的自同构群G的最大数以及在这种情况下G的结构。(4)改进了Alspach和Berggren关于其自同构群具有最大阶的v阶竞赛的结果。也就是说，我们得到了v是3-群或(3，5)-群的充要条件。(5)我们与菲律宾马尼拉的Raposa一起证明了对于RH型的近三正则对称设计，3-交数偶是唯一的。

(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.