show/hide this revision's text 2 tactical configuration

To answer your question about interesting combinatorial objects: Your Sylvester-Hadamard matrix example generalizes in at least two ways.

  1. The incidence matrix of any balanced incomplete block design or, more generally, $t$-design has constant row and column sums. Specifically, a BIBD$(v,b,r,k,\lambda)$ represented as a $v\times b$ incidence matrix has row sums $r$ and column sums $k$. Normalizing an $n\times n$ Hadamard matrix so that the first row and column consist entirely of 1s, and then removing this row and column while replacing $-1$s with 0s gives a design with parameters $v=b=n-1$, $r=k=n/2-1$, $\lambda=n/4-1$. So among Hadamard matrices, Sylvester-Hadamard matrices are not special in this regard. Finite projective planes are also designs, with parameters $v=b=q^2+q+1$, $r=k=q+1$, $\lambda=1$, where $q$ is the order of the plane.

  2. Sylvester-Hadamard matrices have the additional property that if the first row and column are removed, and the remaining rows and columns are suitably permuted, one obtains a circulant matrix. (Most Hadamard matrices do not have this property, but Paley-Hadamard matrices constructed using quadratic residues in $\mathbb{F}_p$, $p\equiv3\pmod{4}$ also do. More generally, Hadamard matrices constructed from difference sets do.) But any $n\times n$ circulant matrix whatsoever will have constant row and column sums (with row sums equal to column sums). (This is not combinatorially so interesting in general.)

Addendum: In answer to your first question, design theorists call an incidence structure whose incidence matrix has constant row and column sums a tactical configuration. This is a $t$-design with $t=1$. The balanced incomplete block designs in the first part of my answer are $t$-designs with $t=2$, but any $t$-design is also a $(t-1)$-design. The condition that any two points be incident with exactly $\lambda$ blocks gives BIBDs a lot of additional interesting structure that tactical configurations do not typically have. 3-, 4-, and 5-designs are more interesting still. See the PlanetMath page for more information.
show/hide this revision's text 1

To answer your question about interesting combinatorial objects: Your Sylvester-Hadamard matrix example generalizes in at least two ways.

  1. The incidence matrix of any balanced incomplete block design or, more generally, $t$-design has constant row and column sums. Specifically, a BIBD$(v,b,r,k,\lambda)$ represented as a $v\times b$ incidence matrix has row sums $r$ and column sums $k$. Normalizing an $n\times n$ Hadamard matrix so that the first row and column consist entirely of 1s, and then removing this row and column while replacing $-1$s with 0s gives a design with parameters $v=b=n-1$, $r=k=n/2-1$, $\lambda=n/4-1$. So among Hadamard matrices, Sylvester-Hadamard matrices are not special in this regard. Finite projective planes are also designs, with parameters $v=b=q^2+q+1$, $r=k=q+1$, $\lambda=1$, where $q$ is the order of the plane.

  2. Sylvester-Hadamard matrices have the additional property that if the first row and column are removed, and the remaining rows and columns are suitably permuted, one obtains a circulant matrix. (Most Hadamard matrices do not have this property, but Paley-Hadamard matrices constructed using quadratic residues in $\mathbb{F}_p$, $p\equiv3\pmod{4}$ also do. More generally, Hadamard matrices constructed from difference sets do.) But any $n\times n$ circulant matrix whatsoever will have constant row and column sums (with row sums equal to column sums). (This is not combinatorially so interesting in general.)