A characterization of some odd sets in projective spaces of order 4 and the extendability of quaternary linear codes

摘要

A set K in PG(r, 4), r ≥ 2, is odd if every line meets K in an odd number of points. An odd set K in PG(r, 4), r ≥ 3, is FH-free if there is no plane meeting K in a Fano plane or in a non-singular Hermitian curve. We prove that an odd set K contains a hyperplane of PG(r, 4) if and only if K is FH-free. As an application to coding theory, a new extension theorem for quaternary linear codes is given.