On nonisomorphic Room squares

Abstract:

Let ${\text {NR}}(s)$ denote the number of nonisomorphic Room squares of side $s$. We prove that for $s$ sufficiently large, ${\text {NR}}(s) \geqslant \exp (c{s^2})$ for some absolute constant $c$. More precisely, ${\text {NR}}(s) \geqslant .19\exp (.04{s^2})$ for $s \geqslant 153\;{\operatorname {odd}}$; and ${\text {NR}}(s) \geqslant .19\exp (.09{s^2})$ for $s \geqslant 1001{\text { odd }}$.