Periodic complementary binary sequences

Since man devised a way to count, patterns or sequences of numbers have held mathematicians\u27 attention. However, it has only been recently that binary sequences have received so much attention due to their compatibility with computers. For the past 30 years the fields of Electrical Engineering, Computer Engineering and Mathematics have done some extensive research in this area because of its many applications in radar, sonar and the reliability and security of communication systems such as digital communications, faxes, telecommunications and electronic transfers especially in banking and finance. First discussed will be the Perfect Binary Sequence. Then the Golay Complementary Sequence and finally the general case of the Periodic Complementary Binary Sequence or PCBS

Algorithms for difference families in finite abelian groups

Our main objective is to show that the computational methods that we previously developed to search for difference families in cyclic groups can be fully extended to the more general case of arbitrary finite abelian groups. In particular the power density PSD-test and the method of compression can be used to help the search.Comment: 18 pages, minor change

Periodic Golay pairs and pairwise balanced designs

In this paper we exploit a relationship between certain pairwise balanced designs with v points and periodic Golay pairs of length v, to classify periodic Golay pairs of length less than 40. In particular, we construct all pairwise balanced designs with v points under specific block conditions having an assumed cyclic automorphism group, and using isomorph rejection which is compatible with equivalence of corresponding periodic Golay pairs, we complete a classification up to equivalence. This is done using the theory of orbit matrices and some compression techniques which apply to complementary sequences. We use similar tools to construct new periodic Golay pairs of lengths greater than 40 where classifications remain incomplete and demonstrate that under some extra conditions on its automorphism group, a periodic Golay pair of length 90 will not exist. Length 90 remains the smallest length for which existence of a periodic Golay pair is undecided. Some quasi-cyclic self-orthogonal codes are constructed as an added application