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

Abstract

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