Hadamard Matrices

In the last chapter of Ian Stewart's book "The Great Mathematical Problems" (ISBN 978-1846683374), there is a section about Hadamard matrices. These are square arrays whose entries are either 1 or 0 (Hadamard's original definition used −1 of 0). Each row in the matrix, when compared with some different row, must have exactly one half of the entries matching and one half non-matching. The first 4 matrices are:

2x2      4x4            8x8                  12x12
1 1    1 1 1 1    1 1 1 1 1 1 1 1    1 1 1 1 1 1 1 1 1 1 1 1
1 0    1 0 1 0    1 0 1 0 1 0 1 0    1 0 0 1 0 1 0 1 0 1 0 1
       1 1 0 0    1 1 0 0 1 1 0 0    1 0 1 0 1 1 0 0 0 0 1 1
       1 0 0 1    1 0 0 1 1 0 0 1    1 1 0 0 1 0 0 1 0 1 1 0
                  1 1 1 1 0 0 0 0    1 0 1 1 1 0 1 1 0 0 0 0
                  1 0 1 0 0 1 0 1    1 1 1 0 0 0 1 0 0 1 0 1
                  1 1 0 0 0 0 1 1    1 0 0 0 1 1 1 0 1 1 0 0
                  1 0 0 1 0 1 1 0    1 1 0 1 1 0 0 0 1 0 0 1
                                     1 0 0 0 0 0 1 1 1 0 1 1
                                     1 1 0 1 0 1 1 0 0 0 1 0
                                     1 0 1 1 0 0 0 0 1 1 1 0
                                     1 1 1 0 0 1 0 1 1 0 0 0

Hadamard matrices have many applications, among them error-correcting codes.

The conjecture

Jacques Hadamard (1865 - 1963) himself proved that, apart from the 2 x 2 matrix, the edge lengths of all Hadamard matrices must be divisible by 4. In 1867 Sylvester gave a construction for all Hadamard matrices of order 2^{k, k >= 2. Paley published two construction methods for two special clases of orders in 1933. The smallest order that could not be constructed by a combination of Sylvester's and Paley's methods was 92. Several other authors published specialized constructions and raised the minimal unknown order to 428, and since 2005 that order is 668.}

Mathematicians believe, but still cannot prove, that Hadamard matrices of all orders 4*n can be constructed.

The 3D model

Since the 0/1-entries of the matrices quickly appear as random noise, I tried to visualize the inner structure of the matrices in a better way, and one result is a 3D model for the 11 matrices with 4, 8, 12, ... 44 = 4*n for n = 1..11.

As one can see in the examples above, there basic building blocks of 2x2 entries. In a first step, the two rows are encoded as binary values:

a b -> a b c d 1 1 -> 1 1 1 0

c d 1 0

Technical details

The article https://arxiv.org/html/2411.18897v1 describes in detail how to compute the matrices with SageMath. Online you can run https://sagecell.sagemath.org/ with the following program:

from sage.combinat.matrices.hadamard_matrix import hadamard_matrix, skew_hadamard_matrix

for i in range(1,11):
  H = hadamard_matrix(4*i)
  print(i, "========")
  print(H.str())