Designs, Codes and Cryptography

, Volume 53, Issue 1, pp 59–68 | Cite as

On the maximality of linear codes

  • T. L. Alderson
  • András Gács


We show that if a linear code admits an extension, then it necessarily admits a linear extension. There are many linear codes that are known to admit no linear extensions. Our result implies that these codes are in fact maximal. We are able to characterize maximal linear (n, k, d) q -codes as complete (weighted) (n, nd)-arcs in PG(k − 1, q). At the same time our results sharply limit the possibilities for constructing long non-linear codes. The central ideas to our approach are the Bruen-Silverman model of linear codes, and some well known results on the theory of directions determined by affine point-sets in PG(k, q).


Codes Non-linear codes Code extension BRS model 

Mathematics Subject Classifications (2000)

94B27 51E20 94B65 


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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of New Brunswick Saint JohnSaint JohnCanada
  2. 2.Department of Computer ScienceEötvös Loránd UniversityBudapestHungary

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