(19, 9, 4) Hadamard designs and their residual designs

In this paper we study (19, 9, 4) Hadamard designs and their residual designs. We prove that there are precisely six non-isomorphic solutions of (19, 9, 4) designs and that these six designs give rise to in all twenty-one mutually non-isomorphic residual designs.     

Hyperovals and Hadamard designs

In this paper, we investigatec-sets in 2-designs, with particular regard to sets of type (0,n) in projective planes. In particular, we associate a Hadamard design to a hyperoval of a projective plane of even orderq and we investigate some properties of its lines. This gives information on the order of the projective plane.     

Nonisomorphic Hadamard designs

A block b of a Hadamard design is called a good block if the symmetric difference b + b₁ is also a block for all nonparallel blocks b₁. The isomorphism classes of such designs having a good block are shown to be related to a double coset decomposition of a symmetric group. As an example, over one million mutually nonisomorphic 3-(32, 16, 7) designs of a certain type are constructed.Equivalence of Hadamard matrices is described in terms of designs and it is shown that nonisomorphic designs may arise from the same matrix.     

Some Hadamard matrices of order 32 and their binary codes

It is known that all doubly‐even self‐dual codes of lengths 8 or 16, and the extended Golay code, can be constructed from some binary Hadamard matrix of orders 8, 16, and 24, respectively. In this note, we demonstrate that every extremal doubly‐even self‐dual [32,16,8] code can be constructed from some binary Hadamard matrix of order 32. © 2004 Wiley Periodicals, Inc.     

On affine designs and Hadamard designs with line spreads

Rahilly [On the line structure of designs, Discrete Math. 92 (1991) 291-303] described a construction that relates any Hadamard design H on 4^m-1 points with a line spread to an affine design having the same parameters as the classical design of points and hyperplanes in AG(m,4). Here it is proved that the affine design is the classical design of points and hyperplanes in AG(m,4) if, and only if, H is the classical design of points and hyperplanes in PG(2m-1,2) and the line spread is of a special type. Computational results about line spreads in PG(5,2) are given. One of the affine designs obtained has the same 2-rank as the design of points and planes in AG(3,4), and provides a counter-example to a conjecture of Hamada [On the p-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its applications to error-correcting codes, Hiroshima Math. J. 3 (1973) 153-226].     

Modular Hadamard martrices and related designs

A square matrix with entries ± 1 is called a modular Hadamard matrix if the inner product of each two distinct row vectors is a multiple of some fixed (positive) integer. This paper initiates the study of modular Hadamard matrices and the combinatorial designs associated with them. The related combinatorial designs are the main concern of this paper; some results dealing with the existence and construction of modular Hadamard matrices will be included in a later paper.     

Switching codes and designs

Various local transformations of combinatorial structures (codes, designs, and related structures) that leave the basic parameters unaltered are here unified under the principle of switching. The purpose of the study is threefold: presentation of the switching principle, unification of earlier results (including a new result for covering codes), and applying switching exhaustively to some common structures with small parameters.     

Unitary designs and codes

A unitary design is a collection of unitary matrices that approximate the entire unitary group, much like a spherical design approximates the entire unit sphere. In this paper, we use irreducible representations of the unitary group to find a general lower bound on the size of a unitary t-design in U(d), for any d and t. We also introduce the notion of a unitary code—a subset of U(d) in which the trace inner product of any pair of matrices is restricted to only a small number of distinct absolute values—and give an upper bound for the size of a code with s inner product values in U(d), for any d and s. These bounds can be strengthened when the particular inner product values that occur in the code or design are known. Finally, we describe some constructions of designs: we give an upper bound on the size of the smallest weighted unitary t-design in U(d), and we catalogue some t-designs that arise from finite groups.      

Structure and automorphism groups of Hadamard designs

Let n be the order of a Hadamard design, and G any finite group. Then there exists many non-isomorphic Hadamard designs of order 2^{12|G| + 13} n with automorphism group isomorphic to G.This research was supported in part by the National Science Foundation.     

Almost cover-free codes and designs

An s-subset of codewords of a binary code X is said to be \((s,\,\ell )\) -bad in X if the code X contains a subset of \(\ell \) other codewords such that the conjunction of the \(\ell \) codewords is covered by the disjunctive sum of the s codewords. Otherwise, the s-subset of codewords of X is called \((s,\,\ell )\) -good in X. A binary code X is said to be a cover-free (CF) \((s,\,\ell )\)-code if the code X does not contain \((s,\,\ell )\)-bad subsets. In this paper, we introduce a natural probabilistic generalization of CF \((s,\,\ell )\)-codes, namely: a binary code X is said to be an almost CF \((s,\,\ell )\)-code if the relative number of its \((s,\,\ell )\)-good s-subsets is close to 1. We develop a random coding method based on the ensemble of binary constant weight codes to obtain lower bounds on the capacity of such codes. Our main result shows that the capacity for almost CF \((s,\,\ell )\)-codes is essentially greater than the rate for ordinary CF \((s,\,\ell )\)-codes.     

Skew quasi-cyclic codes over Galois rings

Recently there has been a lot of interest on algebraic codes in the setting of skew polynomial rings. In this paper we have studied skew quasi-cyclic (QC) codes over Galois rings. We have given a necessary and sufficient condition for skew cyclic codes over Galois rings to be free, and determined a distance bound for free skew cyclic codes. A sufficient condition for 1-generator skew QC codes to be free is determined. Some distance bounds for free 1-generator skew QC codes are discussed. A canonical decomposition of skew QC codes is presented.     

Modular Hadamard matrices and related designs,III

This paper continues the investigations presented in two previous papers on the same subject by the author and A. T. Butson. Modular Hadamard matrices havingn odd andh ≡ ? 1 (modn) are studied for a few values of the parametersn andh. Also, some results are obtained for the two related combinatorial designs. These results include: a discussion on the known techniques for constructing pseudo (v, k, λ)-designs; the fact that the existence of one of the two related designs always implies the existence of the other; and some information about the column sums of the incidence matrix of each of the two ‘maximal’ cases of (m, v, k ₁,λ ₁,k ₂,λ ₂,f, λ ₃)-designs.     

The codes and the lattices of Hadamard matrices

It has been observed by Assmus and Key as a result of the complete classification of Hadamard matrices of order 24, that the extremality of the binary code of a Hadamard matrix H of order 24 is equivalent to the extremality of the ternary code of H^T. In this note, we present two proofs of this fact, neither of which depends on the classification. One is a consequence of a more general result on the minimum weight of the dual of the code of a Hadamard matrix. The other relates the lattices obtained from the binary code and the ternary code. Both proofs are presented in greater generality to include higher orders. In particular, the latter method is also used to show the equivalence of (i) the extremality of the ternary code, (ii) the extremality of the Z₄-code, and (iii) the extremality of a lattice obtained from a Hadamard matrix of order 48.     

Hadamard matrices and doubly even self-dual error-correcting codes

Let n be an integer with n≡4 (mod 8). For any Hadamard matrices H_n of order n, we give a method to define a doubly even self-dual [2n, n] code C(NH_n). Then we will prove that two Hadamard equivalent matrices define equivalent codes.     

D-optimal designs embedded in Hadamard matrices and their effect on the pivot patterns

In this paper we develop a new approach for detecting if specific D-optimal designs exist embedded in Sylvester-Hadamard matrices. Specifically, we investigate the existence of the D-optimal designs of orders 5, 6, 7 and 8. The problem is motivated to explaining why specific values appear as pivot elements when Gaussian elimination with complete pivoting is applied to Hadamard matrices. Using this method and a complete search algorithm we explain, for the first time, the appearance of concrete pivot values for equivalence classes of Hadamard matrices of orders n = 12, 16 and 20.     

Orbit matrices of Hadamard matrices and related codes

In this paper we introduce the notion of orbit matrices of Hadamard matrices with respect to their permutation automorphism groups and show that under certain conditions these orbit matrices yield self-orthogonal codes. As a case study, we construct codes from orbit matrices of some Paley type I and Paley type II Hadamard matrices. In addition, we construct four new symmetric (100,45,20) designs which correspond to regular Hadamard matrices, and construct codes from their orbit matrices. The codes constructed include optimal, near-optimal self-orthogonal and self-dual codes, over finite fields and over ${\mathbb{Z}}_4$.