We consider the defect theorem in the context of labelled polyominoes, i.e., two-dimensional figures. The classical version of this property states that if a set of n words is not a code then the words can be expressed as a product of at most n - 1 words, the smaller set being a code. We survey several two-dimensional extensions exhibiting the boundaries where the theorem fails. In particular, we establish the defect property in the case of three dominoes (n × 1 or 1 × n rectangles).

For polyominoes coded by their boundary word, we describe a quadratic O(n2) algorithm in the boundary length n which improves the naive O(n4) algorithm. Techniques used emanate from algorithmics, discrete geometry and combinatorics on words.