Self-similar carpets over finite fields

In [Dann E. Passoja, Akhlesh Lakhtakia, Carpets and rugs: An exercise in numbers, Leonardo 25 (1) (1992) 69–71] an informal algorithm ‘to display interesting numeric patterns’ is described without any proof. We generalize this algorithm over arbitrary finite fields F q of characteristic p and we prove that it always generates some self-similar sets. For the prime fields F p the generalized algorithm produces p − 1 different self-similar sets. These sets are classified according to their arithmetic and their groups of symmetry.