This page contains the existence and exhaustive results from A search for Hadamard matrices of Williamson type (arXiv:2605.08661).
Definition
Following the definition in the paper: four n × n circulant {-1,1}-matrices A, B, C, D are near-Williamson matrices of order n if they satisfy
AAT + BBT + CCT + DDT = 4nI
and they are all symmetric except possibly for A.
Below is a database of all near-Williamson matrices from the paper. Each file below holds one quadruple found by our search. We extend the known existence of near-Williamson matrices in every odd order up to 67, and provide some additional larger orders. Note that good matrices and Williamson matrices are also near-Williamson matrices. Turyn's infinite family supplies one in every order (q + 1)/2 with q a prime power congruent to 1 modulo 4, which accounts for the orders 79, 91, 97 and 181 shown below. Additionally, good matrices of order 127 were constructed by Djokovic in 1993, so at least those five orders were already known. The new orders below are: 65, 67, 73, 93, 95, 103, 109, 133. (Hadamard matrices of Williamson type were known at orders 73, 93, 95 and 109, but those need not be circulant, so they need not be near-Williamson matrices.) In every quadruple below, no cyclic shift of the first row of A makes A symmetric or skew-type, so none of these are strictly Williamson matrices or good matrices.
Each file contains 4 lines of n entries, the
first rows of A, B, C, D in that order, with
1 = 1 and - = -1.
Below are the exhaustive results from the paper. Apart from the trivial order n = 1, each count links to a file of quadruples of that order: all of them when the order has at most 1000, and the first 1000 as produced by the search otherwise.
| n | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 |
|---|---|---|---|---|---|---|---|---|---|
| # | 1 | 1 | 1 | 3 | 5 | 5 | 24 | 96 | 96 |
| n | 19 | 21 | 23 | 25 | 27 | 29 | 31 | 33 | 35 |
| # | 200 | 1004 | 827 | 2843 | 6579 | 6744 | 11837 | 57003 | 73630 |
| n | 37 | 39 | 41 | 43 | 45 | 47 | |||
| # | 56508 | 206424 | 170197 | 274814 | 1406536 | 654041 |
Encoding: each line of these files is one quadruple, written as
n hexadecimal digits, one digit per position of the four first rows
read from left to right. The digit at position k is
8 A_k + 4 B_k + 2 C_k + 1 D_k, where X_k is 0 when the
entry of X at position k is 1 and is 1 when that entry is -1.
A line for order n
is exactly n digits long, and its first digit is always 0 because each
row is negated, where needed, so that it starts with 1.
Computations were performed on Digital Research Alliance of Canada systems. Parts of the search and of this website were prepared with the assistance of Claude (Anthropic).