Asymptotic Density of Brick and Word Codes

Abstract

Bricks are polyominoes with labelled cells. The problem whether a given set of bricks is a code is undecidable in general. We consider sets consisting of square bricks only. We have shown that in this setting, the codicity of small sets (two bricks) is decidable, but $15$ bricks are enough to make the problem undecidable. Thus the step from words to even simple shapes changes the algorithmic properties significantly (codicity is easily decidable for words). In the present paper we are interested whether this is reflected by quantitative properties of words and bricks. We use their combinatorial properties to show that the proportion of codes among all sets is asymptotically equal to $1$ in both cases.