A method of constructing hadamard matrices of order 𝟐 𝒏+𝟏 (𝒒 + 𝟏) for 𝒒 ≡ 𝟏(𝐦𝐨𝐝 𝟒)

Abstract

A Hadamard matrix 𝐻 of order 𝑛 with entries ±1 satisfying 𝐻𝐻 𝑇 = 𝑛𝐼𝑛, where 𝐻 𝑇 is the transpose of 𝐻 and 𝐼𝑛is the identity matrix of order 𝑛. If 𝐻 = 𝐻 𝑇 , then 𝐻 is called a symmetric Hadamard matrix. The French mathematician Jacques Hadamard proved that such matrices could exist only if 𝑛 is 1, 2, or a multiple of 4. There are many properties and features that define a Hadamard matrix. Two Hadamard matrices are said to be equivalent if one can be obtained from the other by a combination of elementary row operations and column operations. Hadamard matrices can be constructed in many ways such as Sylvester Construction, Paley Construction, Kronecker product construction, Williamson construction etc. In this study, we propose an alternative method to construct inequivalent symmetric Hadamard matrices. A symmetric Hadamard matrix (𝐻2 𝑛₊₁ (𝑞₊₁) ) of order 2 𝑛₊₁ (𝑞 ₊₁) can be constructed by replacing all 0 entries of 𝐻2 𝑛₊₁ (𝑞₊₁) = [ 0 𝑗 𝑇 𝑗 𝑅 ] by the matrix 𝐴2 𝑛₊₁ = [ 𝐴2 𝑛 − 𝐴2 𝑛 −𝐴2 𝑛 − 𝐴2 𝑛 ], where𝐴2 = [ 1 − 1 −1 − 1 ], and all ±1 entries by the matrix ±𝐵2 𝑛₊₁ = ± [ 𝐵2 𝑛 𝐵2 𝑛 𝐵2 𝑛 − 𝐵2 𝑛 ], where 𝐵2 = [ 1 1 1 − 1 ] and 𝑗 is a column vector of length 𝑞 with all entries 1. Here, 𝑞 ≡ 1(mod 4) for a positive integer 𝑛. Moreover, 𝑅 is a Symmetric matrix of order 𝑞 and it is constructed by using 𝜒(𝑎) ̅̅̅̅̅̅, where 𝜒(𝑎) ̅̅̅̅̅̅ = { −1 if 𝑎 is a non zero quadratic residue in 𝐺𝐹(𝑞), 1 if 𝑎 is a quadratic non − residue in 𝐺𝐹(𝑞), 0 if 𝑎 = 0. The quadratic character 𝜒(𝑎) ̅̅̅̅̅̅ indicates whether the given finite field element 𝑎 is a perfect square. If element 𝑎 in 𝐺𝐹(𝑞) is said to be quadratic residue if it is a perfect square in 𝐺𝐹(𝑞) otherwise 𝑎 is a quadratic non-residue. Furthermore, these Hadamard matrices and the Hadamard matrices constructed by using the Sylvester construction are inequivalent. As future work, we plan on implementing a computer programme to construct large inequivalent symmetric Hadamard matrices of order 2 𝑛₊₁ (𝑞 ₊₁).