On the Nonexistence of Perfect Splitter Sets

Splitter sets are closely related to lattice tilings, and have applications in flash memories and conflict avoiding codes. In this paper, we prove some nonexistence results for nonsingular perfect splitter sets. We also give some necessary conditions for the existence of purely singular perfect splitter sets. Finally, we apply these results to purely singular perfect <inline-formula> <tex-math notation="LaTeX">$B[-1,k](m)$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$B[-2,k](m)$ </tex-math></inline-formula> sets for small <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>. In particular, we solve completely the problems left by Schwartz (European J. Combin., vol. 36, pp.130–142, Feb. 2014).

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