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The Metric Structure of Linear Codes

  • Diego Ruano
Chapter

Abstract

The bilinear form with associated identity matrix is used in coding theory to define the dual code of a linear code, also it endows linear codes with a metric space structure. This metric structure was studied for generalized toric codes and a characteristic decomposition was obtained, which led to several applications as the construction of stabilizer quantum codes and LCD codes. In this work, we use the study of bilinear forms over a finite field to give a decomposition of an arbitrary linear code similar to the one obtained for generalized toric codes. Such a decomposition, called the geometric decomposition of a linear code, can be obtained in a constructive way; it allows us to express easily the dual code of a linear code and provides a method to construct stabilizer quantum codes, LCD codes and in some cases, a method to estimate their minimum distance. The proofs for characteristic 2 are different, but they are developed in parallel.

Notes

Acknowledgements

This problem was proposed by Antonio Campillo, I thank him for his many helpful comments. The author gratefully acknowledges the support from RYC-2016-20208 (AEI/FSE/UE), the support from The Danish Council for Independent Research (Grant No. DFF–4002-00367), and the support from the Spanish MINECO/FEDER (Grants No. MTM2015-65764-C2-2-P and MTM2015-69138-REDT).

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Authors and Affiliations

  1. 1.IMUVA (Mathematics Research Institute)University of ValladolidValladolidSpain
  2. 2.Department of Mathematical SciencesAalborg UniversityAalborgDenmark

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