New Codes from Old; A New Geometric Construction

We want to provide some background from coding theory and geometry. Let C be a binary linear code of length N, dimension k, and minimum distance at least 4. Let G be a generator matrix for C of size k_N. Then C has length N and dimension N&k. Put N&k=n+1. A basis for C gives a matrix M of size (n+1)_N. Since C has minimum distance at least 4 it follows that the columns of M form a set S of N points in 7=PG(n, 2) with no 3 collinear. Such a set S with no three of its points collinear is called a cap. Let us say that C is extendable if C can be embedded as a subspace of codimension 1 in a binary linear code D of dimension k+1, length N+1 and minimum distance at least 4. Otherwise C is said to be inextendable or doi:10.1006 jcta.2000.3143, available online at http: www.idealibrary.com on