$\omega\times\omega$-Hadamard matrices

Question

In the following, we define infinite Hadamard matrices.

Let $\omega$ be the set of non-negative integers. If $f,g:\omega\to\{-1,1\}$ are maps, then we say $f,g$ are approximately orthogonal if $$\liminf_{n\to\infty}\sum_{k=0}^n \big(f(k)\cdot g(k)\big)=0.$$

An infinite Hadamard matrix is a map $M:\omega^2 \to \{-1,1\}$ such that whenever $i\neq j \in\omega$ then the "row vectors" $M[i,\cdot]$ and $M[j,\cdot]$ are approximately orthogonal. (Given $a\in\omega$, the map $M[a,\cdot]:\omega\to\{-1,1\}$ is defined by $n\mapsto M(a, n)$.)

Question. What is an example of an infinite Hadamard matrix?

Answer 1

One example would be the universal orthogonal array from the theory of factorial experimental design. The matrix is presented below, followed by its mathematical description.

$\begin{array}{cccccccc} —&{i=0}&1&2&3&4&5&...\\ j=0&1&1&1&1&1&1&...\\ j=1&1&-1&1&-1&1&-1&...\\ j=2&1&1&-1&-1&1&1&...\\ j=3&1&-1&-1&1&1&-1&...\\ j=4&1&1&1&1&-1&-1&...\\ j=5&1&-1&1&-1&-1&1&...\\ ...&...&...&...&...&...&...&...\end{array}$

Let the column number $i$ and row number $j$ be defined as indicated above. Express each as its binary representation according to

$i=\sum\limits_{k=0}^\infty b_{ik}2^k$

$j=\sum\limits_{k=0}^\infty b_{jk}2^k$

$b_{ik},b_{jk}\in\{0,1\}$

(Sums are written to infinity, but for any finite coordinates only finitely many terms are nonzero.)

Then the entry in column $i$, riw $j$ is given by

$\exp_{-1}[\sum\limits_{k=0}^\infty (b_{ik}b_{jk})]$

where $\exp_{-1}$ is use to mean exponential with base $-1$, this nomenclature being used to simplify subscripting. This matrix is exactly orthogonal if it is truncated to $2^n$ rows and $2^n$ columns for any whole number $n$, the resulting truncated arrays then representing the treatments for different factors in a two-level factorial experimental design matrix.