On linear codes whose weights and length have a common divisor

Abstract

In this paper we prove that a set of points (in a projective space over a finite field of q elements), which is incident with 0 mod r points of every hyperplane, has at least ( r − 1 ) q + ( p − 1 ) r points, where 1 < r < q = p h , p prime. An immediate corollary of this theorem is that a linear code whose weights and length have a common divisor r < q and whose dual minimum distance is at least 3, has length at least ( r − 1 ) q + ( p − 1 ) r . The theorem, which is sharp in some cases, is a strong generalisation of an earlier result on the non-existence of maximal arcs in projective planes; the proof involves polynomials over finite fields, and is a streamlined and more transparent version of the earlier one.