Williamson Matrices Research Articles

Hadamard matrices and $\delta $-codes of length $3n$

It is found that four-symbol $\delta$-codes of length $t = 3n$ can be composed for odd $n \leqslant 59$ or $n = {2^a}{10^b}{26^c} + 1$, where all $a$, $b$ and $c \geqslant 0$. Consequently new families of Hadamard matrices of orders $4tw$ and $20tw$ can be constructed, where $w$ is the order of Williamson matrices.

Some infinite classes of Hadamard matrices

Some results of Williamson [ Duke Math. J., 11 1944, Bull. Amer. Math. Soc., 53 1947] and Wallis ( J. Combinatorial Theory, 6 1969] are considerably improved to establish that in each case referred to, the same stated condition or conditions, which according to either of the authors give rise to one Hadamard matrix, actually imply the existence of an infinite series of Hadamard matrices. Also proved is the existence of some infinite series of Williamson's matrices, which coupled with the interesting findings of Turyn [ J. Combinatorial Theory Ser. A, 16 1974] establish the existence of infinitely many more series of Hadamard matrices than those known so far.

An infinite family of Williamson matrices

AbstractIn this paper the following result is proved. Suppose there exists a C-matrix of order n + 1. Then if n≡1 (mod 4) there exists a Hadamard matrix of order 2nr(n + 1), while if n≡3 (mod 4) there exists a Hadamard matrix of order nr(n + 1) for all r ≧0. If n≡1 (mod 4) is a prime power, the method is adapted to prove the existence of a Hadamard matrix of the Williamson type, of order 2nr(n + 1), for all r ≧0.

Construction of Williamson type matrices

Recent advances in the construction of Hadamard matrices have depeaded on the existence of Baumert-Hall arrays and four (1, −1) matrices A B C Dof order m which are of Williamson type, that is they pair-wise satisfy i) MNT = NMT , ∈ {A B C D} and ii) AAT + BBT + CCT + DDT = 4mIm . It is shown that Williamson type matrices exist for the orders m = s(4 − 1)m = s(4s + 3) for s∈ {1, 3, 5, …, 25} and also for m = 93. This gives Williamson matrices for several new orders including 33, 95,189. These results mean there are Hadamard matrices of order i) 4s(4s −1)t, 20s(4s − 1)t,s ∈ {1, 3, 5, …, 25}; ii) 4s(4:s + 3)t, 20s(4s + 3)t s ∈ {1, 3, 5, …, 25}; iii) 4.93t, 20.93t for t ∈ {1, 3, 5, … , 61} ∪ {1 + 2 a 10 b 26 c a b c nonnegative integers}, which are new infinite families. Also, it is shown by considering eight-Williamson-type matrices, that there exist Hadamard matrices of order 4(p + 1)(2p + l)r and 4(p + l)(2p + 5)r when p ≡ 1 (mod 4) is a prime power, 8ris the order of a Plotkin array, and, in the second c...