On the complete pivoting conjecture for a hadamard matrix of order 12 (original) (raw)

On the Complete Pivoting Conjecture for Hadamard Matrices of Small Orders

Journal of Research and Practice in Information Technology

In this paper we study explicitly the pivot structure of Hadamard matrices of small orders 16, 20 and 32. An algorithm computing the (n -j) x (n -j) minors of Hadamard matrices is presented and its implementation for n = 12 is described. Analytical tables summarizing the pivot patterns attained are given.

An Infinite Family of Hadamard Matrices with Fourth Last Pivot n /2

Linear and Multilinear Algebra, 2002

We show that the equivalence class of Sylvester Hadamard matrices give an infinite family of Hadamard matrices in which the fourth last pivot is n/2. Analytical examples of Hadamard matrices of order n having as fourth last pivot n/2 are given for n = 16 and 32. In each case this distinguished case with the fourth pivot n/2 arose in the equivalence class containing the Sylvester Hadamard matrix.

An algorithm to find formulae and values of minors for Hadamard matrices: II

2003

An algorithm computing the (n -j) x (n -j ) , j = 1, 2, ... minors of Hadamard matrices of order n is presented. Its implementation is analytically described step by step for several values of n and j. For j = 7 the values of minors are computed for the first time. A formulae estimating all the values of (n -j) x (n -j) minors is predicted.

Some remarks on Hadamard matrices

Cryptography and Communications, 2010

In this note we use combinatorial methods to show that the unique, up to equivalence, 5 × 5 (1, −1)-matrix with determinant 48, the unique, up to equivalence, 6 × 6 (1,−1)-matrix with determinant 160, and the unique, up to equivalence, 7 × 7 (1,−1)-matrix with determinant 576, all cannot be embedded in the Hadamard matrix of order 8. We also review some properties of Sylvester Hadamard matrices, their Smith Normal Forms, and pivot patterns of Hadamard matrices when Gaussian Elimination with complete pivoting is applied on them. The pivot values which appear reconfirm the above non-embedding results.

A combined approach to the construction of Hadamard matrices

We introduce a general approach which combines the strength of a method recently introduced by Kharaghani with that of orthogonal deorthogonal and sequences, to some very classes of Hadamard matrices. We construct Hadamard matrices of orders 4N(tw+h+k) and 4N(hw+ kv), where v, ware the weights of 2-complementary (0, ±l)-sequenC(~s, and N, t, hand k can take on various values. 'WI, ... , Wu are the weights of the orthogonal design. An equivalent formulation (the usual requires instead that WI, ... , Wn be disjoint (0, ± 1)-matrices and that DDt X1 2 .. + wux})I.

New constructing of regular hadamard matrices

2006

For every prime power q ≡ 7 mod 16, we obtain the (q; a, b, c, d)-partitions of G F (q), with odd integers a, b, c, d, a ≡ ±1 mod 8 such that q = a2 + 2(b2 + c2 + d2) and d2 b2 + 2ac + 2bd. Hence for each value of q the construction of SDS becomes equivalent to building a (q; a, b, c, d)-partition. The latter is much easier than the former. We give a new construction for an infinite family of regular Hadamard matrices of order 4q2 by 16th power cyclotomic classes.

SOME NEW EXAMPLES OF CIRCULANT PARTIAL HADAMARD MATRICES OF TYPE

Advances and Applications in Mathematical Sciences, 2022

Using general possible entries and sign pattern we have forwarded here new examples of Circulant Partial Hadamard matrices of the types   12 6 4   H and  . 28 14 4   H This paper contains basic information about Hadamard matrices, partial Hadamard matrices and circulant partial Hadamard matrices. The recent status of Hadamard matrix conjecture is also included. A table of number of inequivalent circulant partial Hadamard matrices of the type

On Construction of Hadamard Matrices

Iconic Research and Engineering Journals, 2020

This paper describes a specific construction of Hadamard matrices of order with the help of two block matrices and order under some conditions. It is tested that this construction works for =. Some related results are also given.

A construction for Hadamard arrays

Bulletin of the Australian Mathematical Society, 1972

We give a construction for Hadamard arrays and exhibit the arrays of orders 4t, t ∈ {1, 3, 5, 7, …, 19}. This gives seventeen new Hadamard matrices of order less than 4000.