On the Nonexistence of Quaternary [51, 4, 37] Codes

In this paper we prove the nonexistence of quaternary linear codes with parameters [51, 4, 37]. This result gives the exact value ofnq(k,d) forq= 4,k= 4,d= 37 and 38. These were the only minimum distances for which the optimal length of a four-dimensional quaternary code was unknown. The proof is geometrical and relies heavily on results about the structure of certain sets of points inPG(2, 4).