New Codes from Old; A New Geometric Construction

✍ Scribed by Aiden A. Bruen; David L. Wehlau

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 101 KB
• Volume: 94
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.2000.3143

✦ Synopsis

We describe a new technique for obtaining new codes from old ones using geometric methods. Several applications are described.

2001 Academic Press

1. Introduction

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