On a conjecture of Martin on the parameters of completely regular codes and the classification of the completely regular codes in the biggs-smith graph

Abstract

In this paper we study a conjecture of Martin [7] on the paramaters of completely regular codes in distance-regular graphs. We show that this conjecture is not true in general, but for most classical graphs it is true. we also show that there is a counterexample in the Biggs-Smith graph for a weakened version of Martin's conjecture. Furthermore we classify the completely regular codes in the Biggs-Smith graph.