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The Poincaré Polynomial of a Linear Code

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

Author

Listed:
  • Carlos Galindo

    (Universitat Jaume I, Departamento de Matemáticas, Instituto Universitario de Matemáticas y Aplicaciones de Castellón)

  • Fernando Hernando

    (Universitat Jaume I, Departamento de Matemáticas, Instituto Universitario de Matemáticas y Aplicaciones de Castellón)

  • Francisco Monserrat

    (Universidad Politécnica de Valencia, Instituto Universitario de Matemática Pura y Aplicada (IUMPA))

  • Ruud Pellikaan

    (Technische Universiteit Eindhoven, Discrete Mathematics)

Abstract

We introduce the Poincaré polynomial of a linear q-ary code and its relation to the corresponding weight enumerator. The question of whether the Poincaré polynomial is a complete invariant is answered affirmatively for q = 2, 3 and negatively for q ≥ 4. Finally we determine this polynomial for MDS codes and, by means of a recursive formula, for binary Reed-Muller codes.

Suggested Citation

  • Carlos Galindo & Fernando Hernando & Francisco Monserrat & Ruud Pellikaan, 2018. "The Poincaré Polynomial of a Linear Code," Springer Books, in: Gert-Martin Greuel & Luis Narváez Macarro & Sebastià Xambó-Descamps (ed.), Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics, pages 525-535, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-96827-8_23
    DOI: 10.1007/978-3-319-96827-8_23
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