On the construction of optimal linear codes of dimension four

A fundamental problem in coding theory is to find nq(k, d), the minimum length n for which an [n, k, d]q code exists. We show that some q -divisible optimal linear codes of dimension 4 over Fq, which are not of Belov type, can be constructed geometrically using hyperbolic quadrics in PG(3, q). We also construct some new linear codes over Fq with q = 7, 8, which determine n7(4, d) for 31 values of d and n8(4, d) for 40 values of d.