Some orthogonal designs and complex Hadamard matrices by using two Hadamard matrices

We prove that if there exist Hadamard matrices of order h and n divisible by 4 then there exist two disjoint W(1/4hn, 1/8hn), whose sum is a (1, -1) matrix and a complex Hadamard matrix of order 1/4hn, furthermore, if there exists an OD(m; s1, s2,··· ,sl) for even m then there exists an OD(1/4hnm; 1/4hns1, 1/4hns2,···, 1/4hnsl)

Some Orthogonal Designs and complex Hadamard matrices by using two Hadamard matrices

We prove that if there exist Hadamard matrices of order h and n divisible by 4 then there exist two disjoint W ( 1 4 hn; 1 8 hn), whose sum is a (1; \Gamma1) matrix and a complex Hadamard matrix of order 1 4 hn, furthermore, if there exists an OD(m; s 1 ; s 2 ; \Delta \Delta \Delta ; s l ) for even m then there exists an OD( 1 4 hnm; 1 4 hns 1 ; 1 4 hns 2 ; \Delta \Delta \Delta ; 1 4 hns l ): 1 Introduction and Basic Definitions A complex Hadamard matrix (see [4] ), say C, of order c is a matrix with elements 1; \Gamma1; i; \Gammai satisfying CC = cI , where C is the Hermitian conjugate of C. From [4], any complex Hadamard matrix has order 1 or order divisible by 2. Let C = X + iY , where X; Y consist of 1; \Gamma1; 0 and X Y = 0 where is the Hadamard product. Clearly, if C is an complex Hadamard matrix then XX T + Y Y T = cI, XY T = Y X T . A weighing matrix [2] of order n with weight k, denoted by W = W (n; k), is a (1; \Gamma1; 0) matrix satisfying WW T ..

Properties of trace maps and their applications to coding theory

In this thesis we study the application of trace maps over Galois fields and Galois rings in the construction of non-binary linear and non-linear codes and mutually unbiased bases. Properties of the trace map over the Galois fields and Galois rings has been used very successfully in the construction of cocyclic Hadamard, complex Hadamard and Butson Hadamard matrices and consequently to construct linear codes over integers modulo prime and prime powers. These results provide motivation to extend this work to construct codes over integers modulo . The prime factorization of integers paved the way to focus our attention on the direct product of Galois rings and Galois fields of the same degree. We define a new map over the direct product of Galois rings and Galois fields by using the usual trace maps. We study the fundamental properties of the this map and notice that these are very similar to that of the trace map over Galois rings and Galois fields. As such this map called the trace-like map and is used to construct cocyclic Butson Hadamard matrices and consequently to construct linear codes over integers modulo . We notice that the codes construct in this way over the integers modulo 6 is simplex code of type . A further generalization of the trace-like map called the weighted-trace map is defined over the direct product of Galois rings and Galois fields of different degrees. We use the weighted-trace map to construct some non-linear codes and mutually unbiased bases of odd integer dimensions. Further more we study the distribution of over the Galois fields of degree 2 and use it to construct 2-dimensional, two-weight, self-orthogonal codes and constant weight codes over integers modulo prime