Wolf H. Holzmann

Complex Golay sequences: structure and applications

Abstract Complex Golay sequences were introduced in 1992 to generalize constructions for Hadamard matrices using Golay sequences. (In the last section of this paper we describe some independent earlier work on quadriphase pairs–equivalent objects used in the setting of signal processing.) Since then we have constructed some new infinite classes of these sequences and learned some facts about their structure. In particular, if the length of complex Golay sequences is divisible by a prime p≡3 mod 4 , then their Hall polynomials have a nontrivial factorization h(x)k(x), cx d h(x)k ∗ (x) as polynomials over GF(p2), where c=a+bi, a 2 +b 2 ≡−1 mod p and k ∗ is obtained from k by a natural involution acting on complex Laurent polynomials. We explain how these facts can be used to simplify the search for complex Golay sequences, and show how to construct a large variety of sets of four complex sequences with zero autocorrelation, suitable for the construction of various matrices such as Hadamard matrices, complex Hadamard matrices and signed group Hadamard matrices over the dihedral signed group.

Extremal Ternary Self-Dual Codes Constructed from Negacirculant Matrices

In this paper, a construction of ternary self-dual codes based on negacirculant matrices is given. As an application, we construct new extremal ternary self-dual codes of lengths 32, 40, 44, 52 and 56. Our approach regenerates all the known extremal self-dual codes of lengths 36, 48, 52 and 64. New extremal ternary quasi-twisted self-dual codes are also constructed.

On the orthogonal designs of order 40

Abstract Using a new method we construct all 17 remaining (unresolved for over 20 years) full orthogonal designs of order 40 in three variables. This implies that all full orthogonal designs OD(2 t 5; x , y , 2 t 5− x − y ) exist for all t ⩾3. The last two remaining orthogonal designs of order 40 in 2 variables are obtained as a special case of two of these designs.

On the orthogonal designs of order 24

Using a new method we construct all 18 remaining (unresolved for 20 years) full orthogonal designs of order 24 in four variables. The last three remaining orthogonal designs of order 24 in 3-variables are obtained as a special case of three of these designs.

A D-optimal design of order 150

Abstract A non-circular Ehlich D-optimal design of order 150 is constructed. Noting that all computer computations so far have not produced a circular D-optimal design of order 150 makes this design quite interesting.

Sequences and arrays involving free variables

Abstact: Sequences in free variables are introduced and used to construct arrays in free variables which are suitable for circulant matrices. Most of the arrays found are maximal in the number of free variables. Applications include many new Goethals-Seidel type arrays and complex orthogonal designs.

The excess problem and some excess inequivalent matrices of order 32

Abstract Let 4 n be the order of an Hadamard matrix. It is shown that there is a regular complex Hadamard matrix of order 8 n 2 . Five classes of excess-inequivalent Hadamard matrices of order 32 are introduced.

All triples for orthogonal designs of order 40

The use of amicable sets of eight circulant matrices and four negacyclic matrices to generate orthogonal designs is relatively new. We find all 1841 possible orthogonal designs of order 40 in three variables, using only these new techniques.