Are these two methods for constructing Hadamard matrices known?

Question

These two observations came while researching the empty set of odd perfect numbers and unitary perfect numbers:

Context:

Let $n$ be a natural number and $D_n$ be the set of divisors. We can make this set to a ring by observing that each divisor $d$ has

$$0 \le v_p(d) \le v_p(n)$$

Hence we can add two divisors $d,e$ by setting:

$$d \oplus e := \prod_{p | n} p^{v_p(d)+v_p(e) \mod (v_p(n)+1)}$$

and similarily we can multiply them by setting: $$d \otimes e := \prod_{p | n} p^{v_p(d) \cdot v_p(e) \mod (v_p(n)+1)}$$

Then, if $n = p_1^{a_1} \cdots p_r^{a_r}$, this ring will be isomorphic to the ring

$$\mathbb{Z}/(a_1+1) \times \cdots \times \mathbb{Z}/(a_r+1)$$

If $n$ is squarefree, than this reduces to :

$$d\oplus e = \frac{de}{\gcd(d,e)^2}$$

$$d\otimes e = \gcd(d,e)$$

Both methods rely on the character tables of abelian groups of order $2^r$ and on Dedekind group matrices defined over the set of divisors or unitary divisors of $n$:

Method one:

Let $U(n):=$ set of unitary ($\gcd(d,n/d)=1$) divisors of $n$ ordered by their absolute value:

Then $H_n$ is a Hadamard matrix:

$$H_n := ((-1)^{\omega(\gcd(d,e))})_{d,e \in U(n)}$$

Method two:

Let $X(n):= \{ \sqrt{\operatorname{rad}(d)d} : d|n, \gcd(d,n/d)=1, \forall p|d: v_p(d)\equiv 1 \mod(2) \}$.

and let $\chi_n(d,e) := \prod_{p|n} \exp(\frac{2 \pi \sqrt{-1}}{v_p(n)+1})^{v_p(d)v_p(e)}$ Then this matrix is a Hadamard matrix as the character table of some abelian group $\mathbb{Z}/(2)^r$:

$$H^{(2)}_n := (\chi_n(d,e))_{d,e \in X(n)}$$

Example computations, show, that these two construction are in general not the same and they are not the same as the induction construction given at Wikipedia:

SageMath-Computations

Are these constructions known or are they maybe equivalent in some sense to the inductive Sylvester construction given at Wikipedia?

Thanks for your help!

Answer 1

I have a feeling that this specific construction is already known but have no time to do an extensive check. There are a number of ways to start a search.

The book Hadamard Matrices and Their Applications by Horadam is a good reference. There is also an online library of known Hadamard matrices maintained by Neil Sloane.

PS: The idea of group developed Hadamard matrices is defined on page 6 of Padraic O'Cathain's master's thesis. The rest of the thesis may be interesting as well.

Answer 2

Two matrices are Hadamard equivalent if they differ from one another by permuting and negating rows and columns. It is known that there is a unique Hadamard equivalence class of Hadamard matrices at orders 2, 4, 8. There are 5 equivalence classes at order 16, and the number grows rapidly beyond that (the number of equivalence classes at order 32 is above 13 million and they were enumerated only in 2013). There are routines implemented in MAGMA (and possibly in other computer algebra systems) for testing Hadamard equivalence.

I think it very probably that your matrices are Hadamard equivalent to the Sylvester matrices. In Method 1, it looks like the divisors of $n$ could be replaced by binary strings of length the number of distinct prime divisors. Then this looks a lot like the construction of Sylvester matrices given in Section 2.1.1 of Horadam's Hadamard matrices and their applications.

In Method 2, you observe that this is the character table of an elementary abelian 2-group. This is the definition I would give for the Sylvester Hadamard matrices. The bijection to Horadam's definition is not too hard to work out: identify the elements of the 2-group with row-vectors of a vector space over $\mathbb{F}_{2}$ and the characters with column vectors.