Weak amicable T‐matrices and Plotkin arrays

We introduce the class of weak amicable T‐matrices and use it to construct a class of orthogonal designs, for p = 1 and for p a prime power ≡ 3 (mod 4), and all odd q, q ≤ 21. This class includes new Plotkin arrays of order 24, 40, 56 and for the first time, of orders 8q, q ∈ {9,11,13,15,17,19,21}. © 2006 Wiley Periodicals, Inc. J Combin Designs 16: 44–52, 2008     

New orthogonal designs of order 56

Amicable sets of eight matrices are very useful in the construction of orthogonal designs using the Kharaghani array. In this article we use a simple procedure to construct many new amicable sets of eight matrices of order 7 and then new orthogonal designs of order 56. Some of these are restricted to be short amicable sets of two or four circulant matrices.. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 387–393, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10030     

Orthogonal designs,self‐dual codes,and the Leech lattice

Symmetric designs and Hadamard matrices are used to construct binary and ternary self‐dual codes. Orthogonal designs are shown to be useful in construction of self‐dual codes over large fields. In this paper, we first introduce a new array of order 12, which is suitable for any set of four amicable circulant matrices. We apply some orthogonal designs of order 12 to construct new self‐dual codes over large finite fields, which lead us to the odd Leech lattice by Construction A. © 2005 Wiley Periodicals, Inc. J Combin Designs 13: 184–194, 2005.     

Symmetric Weighing Matrices Constructed using Group Matrices

A weighing matrix of order n and weight m² is a square matrix M of order n with entries from {-1,0,+1} such that MM^T=m²I where I is the identity matrix of order n. If M is a group matrix constructed using a group of order n, M is called a group weighing matrix. Recently, group weighing matrices were studied intensively, especially when the groups are cyclic and abelian. In this paper, we study the abelian group weighing matrices that are symmetric, i.e.M^T=M. Some new examples are found. Also we obtain a few exponent bounds on abelian groups that admit symmetric group weighing matrices. In particular, we prove that there is no symmetric abelian group weighing matrices of order 2p^r and weight p² where p is a prime and p≥ 5.Communicated by: K.T. Arasu     

A weak difference set construction for higher dimensional designs

We continue the analysis of de Launey's modification of development of designs modulo a finite groupH by the action of an abelian extension function (AEF), and of the proper higher dimensional designs which result.We extend the characterization of allAEFs from the cyclic group case to the case whereH is an arbitrary finite abelian group.We prove that ourn-dimensional designs have the form (f(j ₁ j ₂ ...j _n )) (j _i J), whereJ is a subset of cardinality |H| of an extension groupE ofH. We say these designs have a weak difference set construction.We show that two well-known constructions for orthogonal designs fit this development scheme and hence exhibit families of such Hadamard matrices, weighing matrices and orthogonal designs of orderv for which |E|=2v. In particular, we construct proper higher dimensional Hadamard matrices for all orders 4t100, and conference matrices of orderq+1 whereq is an odd prime power. We conjecture that such Hadamard matrices exist for all ordersv0 mod 4.     

The p-Rank of the Incidence Matrix of Intersecting Linear Subspaces

Let V be a vector space of dimension n+1 over a field of p ^t elements. A d-dimensional subspace and an e-dimensional subspace are considered to be incident if their intersection is not the zero subspace. The rank of these incidence matrices, modulo p, are computed for all n, d, e and t. This result generalizes the well-known formula of Hamada for the incidence matrices between points and subspaces of given dimensions in a finite projective space. A generating function for these ranks as t varies, keeping n, d and e fixed, is also given. In the special case where the dimensions are complementary, i.e., d+e=n+ 1, our formula improves previous upper bounds on the size of partial m-systems (as defined by Shult and Thas).     

On the maximum cocliques of the rank 3 graph of 211 : M24

In this article, we consider the maximum cocliques of the 2¹¹: M₂₄ ‐graph Λ. We show that the maximum cocliques of size 24 of Λ can be obtained from two Hadamard matrices of size 24, and that there are exactly two maximum cocliques up to equivalence. We verify that the two nonisomorphic designs with parameters 5‐(24,9,6) can be constructed from the maximum cocliques of Λ, and that these designs are isomorphic to the support designs of minimum weights of the ternary extended quadratic residue and Pless symmetry [24,12,9] codes. Further, we give a new construction of Λ from these 5‐(24,9,6) designs. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 323–332, 2009     

Embeddings of Un(q2) and symmetric strongly regular graphs

In the geometric setting of the embedding of the unitary group U_n(q²) inside an orthogonal or a symplectic group over the subfield GF(q) of GF(q²), q odd, we show the existence of infinite families of transitive two‐character sets with respect to hyperplanes that in turn define new symmetric strongly regular graphs and two‐weight codes. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 248–253, 2010     

Distribution of Subdominant Eigenvalues of Random Matrices

We mainly investigate the behavior of the subdominant eigenvalue of matrices B= (b _i,j)^n,n whose entries are independent random variables with an expectation Eb _i,j=1/n and with a variance n c/n ² for some constant c 0. For such matrices we show that for large n, the subdominant eigenvalue is, with great probability, in a small neighborhood of 0. We also show that for large n, the spectral radius of such matrices is, with great probability, in a small neighborhood of 1.     

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. © 2000 John Wiley & Sons, Inc. J Combin Designs 9: 17–27, 2001     

Permutation Matrices, Wreath Products, and the Distribution of Eigenvalues

We consider a class of random matrix ensembles which can be constructed from the random permutation matrices by replacing the nonzero entries of the n×n permutation matrix matrix with M×M diagonal matrices whose entries are random Kth roots of unity or random points on the unit circle. Let X be the number of eigenvalues lying in a specified arc I of the unit circle, and consider the standardized random variable (X–E[X])/(Var(X))^1/2. We show that for a fixed set of arcs I ₁,...,I _N , the corresponding standardized random variables are jointly normal in the large n limit, and compare the covariance structures which arise with results for other random matrix ensembles.     

Some cyclic codes of length 2<Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Explicit expressions for 4n + 2 primitive idempotents in the semi-simple group ring $R_{2p^{n}}\equiv \frac{GF(q)[x]}{p and q are distinct odd primes; n ≥ 1 is an integer and q has order \fracf(2pⁿ)2{\frac{\phi(2p^{n})}{2}} modulo 2p ⁿ . The generator polynomials, the dimension, the minimum distance of the minimal cyclic codes of length 2p ⁿ generated by these 4n + 2 primitive idempotents are discussed. For n = 1, the properties of some (2p, p) cyclic codes, containing the above minimal cyclic codes are analyzed in particular. The minimum weight of some subset of each of these (2p, p) codes are observed to satisfy a square root bound.     

Structural properties and recognition of restricted and strongly unimodular matrices

A (0, ±1) matrix A is restricted unimodular if every matrix obtained from A by setting to zero any subset of its entries is totally unimodular. Restricted unimodular matrices are also known as matrices without odd cycles. They have been studied by Commoner and recently Yannakakis has given a polynomial algorithm to recognize when a matrix belongs to this class. A matrix A is strongly unimodular if any matrix obtained from A by setting at most one of its entries to zero is totally unimodular. Crama et al. have shown that (0,1) matrix A is strongly unimodular if and only if any basis of (A, 1) is triangular, whereI is an identity matrix of suitable dimensions. In this paper we give a very simple algorithm to test whether a matrix is restricted unimodular and we show that all strongly unimodular matrices can be obtained by composing restricted unimodular matrices with a simple operation. Partially supported by a New York University Research Challenge Fund Grant.     

Minimum permanents of doubly stochastic matrices with at least one zero entry †

It is shown that the minimum value of the permanent on the n× ndoubly stochastic matrices which contain at least one zero entry is achieved at those matrices nearest to J_n in Euclidean norm, where J_n is the n× nmatrix each of whose entries is n^-1 . In case n ≠ 3 the minimum permanent is achieved only at those matrices nearest J_n ; for n= 3 it is achieved at other matrices containing one or more zero entries as well.     

Automorphisms of higher‐dimensional Hadamard matrices

This article derives from first principles a definition of equivalence for higher‐dimensional Hadamard matrices and thereby a definition of the automorphism group for higher‐dimensional Hadamard matrices. Our procedure is quite general and could be applied to other kinds of designs for which there are no established definitions for equivalence or automorphism. Given a two‐dimensional Hadamard matrix H of order ν, there is a Product Construction which gives an order ν proper n‐dimensional Hadamard matrix P⁽ⁿ⁾(H). We apply our ideas to the matrices P⁽ⁿ⁾(H). We prove that there is a constant c > 1 such that any Hadamard matrix H of order ν > 2 gives rise via the Product Construction to cν inequivalent proper three‐dimensional Hadamard matrices of order ν. This corrects an erroneous assertion made in the literature that ”P⁽ⁿ⁾(H) is equivalent to “P⁽ⁿ⁾(H′) whenever H is equivalent to H′.” We also show how the automorphism group of P⁽ⁿ⁾(H) depends on the structure of the automorphism group of H. As an application of the above ideas, we determine the automorphism group of P⁽ⁿ⁾(Hk) when Hk is a Sylvester Hadamard matrix of order 2^k. For ν = 4, we exhibit three distinct families of inequivalent Product Construction matrices P⁽ⁿ⁾(H) where H is equivalent to H₂. These matrices each have large but non‐isomorphic automorphism groups. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 507–544, 2008     

Williamson matrices up to order 59

A recent result of Schmidt has brought Williamson matrices back into the spotlight. In this article, a new algorithm is introduced to search for hard to find Williamson matrices. We find all nonequivalent Williamson matrices of odd order n up to n = 59. It turns out that there are none for n = 35, 47, 53, 59 and it seems that the Turyn class may be the only infinite class of these matrices.      

Hadamard matrices of small order and Yang conjecture

We show that 138 odd values of n<10000 for which a Hadamard matrix of order 4n exists have been overlooked in the recent handbook of combinatorial designs. There are four additional odd n=191, 5767, 7081, 8249 in that range for which Hadamard matrices of order 4n exist. There is a unique equivalence class of near‐normal sequences NN(36), and the same is true for NN(38) and NN(40). This means that the Yang conjecture on the existence of near‐normal sequences NN(n) has been verified for all even n⩽40, but it still remains open. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 254–259, 2010     

Stable Laws and Products of Positive Random Matrices

Let S be the multiplicative semigroup of q×q matrices with positive entries such that every row and every column contains a strictly positive element. Denote by (X _n ) _n≥1 a sequence of independent identically distributed random variables in S and by X ⁽ⁿ⁾=X _n ⋅⋅⋅ X ₁,  n≥1, the associated left random walk on S. We assume that (X _n ) _n≥1 satisfies the contraction property

A technique for constructing non‐embeddable quasi‐residual designs

We propose a technique for constructing two infinite families of non‐embeddable quasi‐residual designs as soon as one such design satisfying certain conditions exists. The main tools are generalized Hadamard matrices and balanced generalized weighing matrices. Starting with a specific non‐embeddable quasi‐residual 2‐(27,9,4) design, we construct for every positive integer m a non‐embeddable 2‐(3^m,3^m?1,(3^m?1?1)/2)‐design, and, if r_m=(3^m?1)/2 is a prime power, we construct for every positive integer n a non‐embeddable design. For each design in these families, a symmetric design with the corresponding parameters is known to exist. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 160–172, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.900     

Constructing integral matrices with given line sums

D. Gale, in 1957 and H.J. Ryser, in 1963, independently proved the famous Gale–Ryser theorem on the existence of (0, 1)–matrices with prescribed row and column sums. Around the same time, in 1968, Mirsky solved the more general problem of finding conditions for the existence of a nonnegative integral matrix with entries less than or equal to p and prescribed row and column sums. Using the results of Mirsky, Brualdi shows that a modified version of the domination condition of Gale–Ryser is still necessary and sufficient for the existence of a matrix under the same constraints. In this article we prove another extension of Gale–Ryser’s domination condition. Furthermore we present a method to build nonnegative integral matrices with entries less than or equal to p and prescribed row and column sums.