Groups of finite elementary codes (Q685448)

The authors solve a problem posed in \textit{G. Lallement} and \textit{C. Reis} [Inform. Control 48, 11-29 (1981; Zbl 0459.68039)]: Determine all groups of finite indecomposable elementary codes. The authors do even better and give an algorithm to compute the group of any given finite elementary code. Throughout \(A\) will denote a finite alphabet of at least 2 letters and \(A^*\) is the free monoid on \(A\). Theorem 1: A subset \(C\) of \(A^*\) is a finite elementary code iff \(C\) is accepted by a team tournament. Theorem 2: A finite elementary code is indecomposable iff its group is primitive. The authors determine all finite elementary codes whose groups are \(PGL_{2,5}\). Theorem 3: Let \(C\) be a finite indecomposable elementary code of degree \(n\) over \(A\) where \(A\) is of cardinality \(m\). Then one of the following holds: 1) \(n\) is prime or \(n = 1\), and the group of \(C\) is cyclic of order \(n\), 2) \(n \geq 3\) and the group of \(C\) is the symmetric group on \(n\) elements, 3) \(n\) is odd, \(n \geq 5\) and the group of \(C\) is the alternating group on \(n\) elements, 4) \(n = 6\) and the group of \(C\) is \(PGL_{2,5}\). The authors describe the connection between the groups of elementary codes and the groups of their indecomposable components.