Pseudogeometries with clusters and an example of a recursive $[4,2,3]_{42}$-code

In 1998, E. Couselo, S. Gonzalez, V. Markov, and A. Nechaev defined the recursive codes and obtained some results that allowed one to conjecture the existence of recursive MDS-codes of dimension 2 and length 4 over any finite alphabet of cardinality $q\notin\{2,6\}$. This conjecture remained open only for $q\in\{14,18,26,42\}$. It is shown in this paper that there exist such codes for $q=42$. We used a new construction, that of pseudogeometry with clusters.