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Designs, Codes and Cryptography

, Volume 52, Issue 3, pp 363–380 | Cite as

A generalized Gleason–Pierce–Ward theorem

  • Jon-Lark Kim
  • Xiaoyu Liu
Article

Abstract

The Gleason–Pierce–Ward theorem gives constraints on the divisor and field size of a linear divisible code over a finite field whose dimension is half of the code length. This result is a departure point for the study of self-dual codes. In recent years, additive codes have been studied intensively because of their use in additive quantum codes. In this work, we generalize the Gleason–Pierce–Ward theorem on linear codes over GF(q), q = p m , to additive codes over GF(q). The first step of our proof is an application of a generalized upper bound on the dimension of a divisible code determined by its weight spectrum. The bound is proved by Ward for linear codes over GF(q), and is generalized by Liu to any code as long as the MacWilliams identities are satisfied. The trace map and an analogous homomorphism \({x\mapsto x-x^p}\) on GF(q) are used to complete our proof.

Keywords

Additive codes Gleason–Pierce–Ward theorem Divisible codes Ward’s bound 

Mathematics Subject Classification (2000)

94B05 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of LouisvilleLouisvilleUSA
  2. 2.Department of Mathematics and StatisticsWright State UniversityDaytonUSA

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