Quadratic Algorithms for Testing of Codes and ◊-Codes

Abstract

The Sardinas-Patterson's test for codes has contributed many effective testing algorithms to the development of theory of codes, formal languages, etc. However, we will show that a modification of this test proposed in this paper can deduce more effective testing algorithms for codes. As a consequence, we establish a quadratic algorithm that, given as input a regular language X defined by a tuple Φ, M, B, where Φ : A* → M is a monoid morphism saturating X, M is a finite monoid, $B \subseteq M$, X = Φ-1B, decides in time complexity $\mathcal{O}n^2$ whether X is a code, where n = CardM. Specially, n can be chosen as the finite index of X. A quadratic algorithm for testing of ◊-codes is also established.