Maximal number of constant weight vertices of the unit n-cube contained in a k-dimensional subspace

Abstract

Summary form only given. We introduce and solve a seemingly basic geometrical extremal problem. The set of vertices of weight w in the unit cube of R/sup n/, E(n,w)={x/sup n//spl isin/{0,1}/sup n/:x/sup n/ has w ones} can also be viewed as the set in which constant weight codes are studied in information theory. Another interest there is in linear codes. This was a motivation for studying the interplay between two properties: constant weight and linearity. In particular we wanted to know M(n,k,w)/sup /spl Delta///sub =/max{|U/sub k//sup n//spl cap/E(n,w)|: U/sub k//sup n/ is a k-dimensional linear subspace of R/sup n/}, that is, the maximal cardinality of a set of vectors in E(n,w), whose linear span has a dimension not exceeding k. Our complete solution is given. We also present an extension to multi-sets and explain a connection to the (simpler) Erdos-Moser problem.