How many figure sets are codes?

Abstract

Defect theorem, which provides a kind of dimension property for words, does not hold for two-dimensional figures (labelled polyominoes), except for some small sets. We thus turn to the analysis of asymptotic density of figure codes. Interestingly, it can often be proved to be 1, even in those cases where the defect theorem fails. Hence it reveals another weak dimension property which does hold for figures, i.e., non-codes are rare. We show that the asymptotic densities of codes among the following sets are all equal to 1: (ordinary) words, square figures and small sets of dominoes, where small refers to cardinality ≤3. The latter is a borderline case for the defect theorem and additionally exhibits interesting properties at different alphabet sizes. © 2008 Springer-Verlag Berlin Heidelberg.