On Skew E–W Matrices

Abstract

An E–W matrix M is a ( − 1, 1)‐matrix of order 4t+2, where t is a positive integer, satisfying that the absolute value of its determinant attains Ehlich–Wojtas' bound. M is said to be of skew type (or simply skew) if M−I is skew‐symmetric where I is the identity matrix. In this paper, we draw a parallel between skew E–W matrices and skew Hadamard matrices concerning a question about the maximal determinant. As a consequence, a problem posted on Cameron's website [7] has been partially solved. Finally, codes constructed from skew E–W matrices are presented. A necessary and sufficient condition for these codes to be self‐dual is given, and examples are provided for lengths up to 52.

On the Smith normal form of skew E-W matrices

“…[11] We can always assume without loss of generality that every skew E-W matrix is of the form (4). Examples of skew E-W matrices for small orders have been provided in [9].…”

“…However, it was proved [9] that skew E-W matrices may only exist when 2n − 3 = α 2 for some integer α (i.e. β = 1), a condition which is believed to be sufficient.…”

“…β = 1), a condition which is believed to be sufficient. In [9] examples of skew E-W matrices for small orders have been provided.…”

On the Smith normal form of a skew‐symmetric d‐optimal design of order

“…It is well known that must be a sum of squares . Furthermore, Armario and Frau showed that must be a square if is skew‐symmetric. Indeed, they showed the following:…”

“…A {1, −1}-matrix B of order n is called an EW matrix if it satisfies (2). Hence, in particular, an EW matrix of order n has determinant…”