Blocking sets in finite projective spaces and uneven binary codes

Abstract

A 1- blocking set in the projective space PG( m,2), m⩾2, is a set B of points such that any ( m−1)-flat meets B and no 1-flat is contained in B. A binary linear code is said to be uneven if it contains at least one codeword of odd weight. If B is a 1-blocking set in PG( r−1,2) and dim〈 B〉= r−1 any matrix H whose columns are the vectors in B is a parity check matrix for an uneven binary code of length n=| B|, redundancy r, and minimum distance at least 4; Conversely, if B is the set of columns of the parity check matrix of such a code then it is a 1-blocking set. Using this and results on uneven binary codes of minimum distance 4, the author shows that there exists a 1-blocking set of cardinality n if and only if 5⩽ n⩽5⋯2 m−3 .