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The Metric Structure of Linear Codes

In: Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics

Author

Listed:
  • Diego Ruano

    (University of Valladolid, IMUVA (Mathematics Research Institute)
    Aalborg University, Department of Mathematical Sciences)

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.

Suggested Citation

  • Diego Ruano, 2018. "The Metric Structure of Linear Codes," Springer Books, in: Gert-Martin Greuel & Luis Narváez Macarro & Sebastià Xambó-Descamps (ed.), Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics, pages 537-561, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-96827-8_24
    DOI: 10.1007/978-3-319-96827-8_24
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