Discrete Mathematics ( IF 0.770 ) Pub Date : 2019-12-23 , DOI: 10.1016/j.disc.2019.111744
Martin E. Malandro; Ken W. Smith

Let ${G}_{n}$ denote the group ${C}_{{2}^{n}}×{C}_{{2}^{n}}$, where ${C}_{k}$ is the cyclic group of order $k$. We give an algorithm for enumerating the regular nontrivial partial difference sets (PDS) in ${G}_{n}$. We use our algorithm to obtain all of these PDS in ${G}_{n}$ for $2\le n\le 9$, and we obtain partial results for $n=10$ and $n=11$. Most of these PDS are new. For $n\le 4$ we also identify group-inequivalent PDS. Our approach involves constructing tree diagrams and canonical colorings of these diagrams. Both the total number and the number of group-inequivalent PDS in ${G}_{n}$ appear to grow super-exponentially in $n$. For $n=9$, a typical canonical coloring represents in excess of $1{0}^{146}$ group-inequivalent PDS, and there are precisely ${2}^{520}$ reversible Hadamard difference sets.

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