The Poincaré polynomial of a linear code

Carlos Galindo; Fernando  Hernando; Franciso Montserrat; G.R. Pellikaan

doi:10.1007/978-3-319-96827-8_23

The Poincaré polynomial of a linear code

Carlos Galindo, Fernando Hernando, Franciso Montserrat, G.R. Pellikaan

Research output: Chapter in Book/Report/Conference proceeding › Chapter › Academic › peer-review

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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.

Original language	English
Title of host publication	Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics
Subtitle of host publication	Festschrift for Antonio Campillo on the Occasion of his 65th Birthday
Editors	Gert-Martin Greuel, Luis Narváez Macarro, Sebastià Xambó-Descamps
Place of Publication	Cham
Publisher	Springer
Chapter	23
Pages	525-535
Number of pages	11
ISBN (Electronic)	978-3-319-96826-1
ISBN (Print)	978-3-319-96827-8
DOIs	https://doi.org/10.1007/978-3-319-96827-8_23
Publication status	Published - 18 Sept 2018

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Galindo, C., Hernando, F., Montserrat, F., & Pellikaan, G. R. (2018). The Poincaré polynomial of a linear code. In G.-M. Greuel, L. Narváez Macarro, & S. Xambó-Descamps (Eds.), Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics: Festschrift for Antonio Campillo on the Occasion of his 65th Birthday (pp. 525-535). Springer. https://doi.org/10.1007/978-3-319-96827-8_23

Galindo, Carlos ; Hernando, Fernando ; Montserrat, Franciso et al. / The Poincaré polynomial of a linear code. Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics: Festschrift for Antonio Campillo on the Occasion of his 65th Birthday. editor / Gert-Martin Greuel ; Luis Narváez Macarro ; Sebastià Xambó-Descamps. Cham : Springer, 2018. pp. 525-535

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title = "The Poincar{\'e} polynomial of a linear code",

abstract = "We introduce the Poincar{\'e} polynomial of a linear q-ary code and its relation to the corresponding weight enumerator. The question of whether the Poincar{\'e} 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.",

author = "Carlos Galindo and Fernando Hernando and Franciso Montserrat and G.R. Pellikaan",

year = "2018",

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doi = "10.1007/978-3-319-96827-8_23",

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Galindo, C, Hernando, F, Montserrat, F & Pellikaan, GR 2018, The Poincaré polynomial of a linear code. in G-M Greuel, L Narváez Macarro & S Xambó-Descamps (eds), Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics: Festschrift for Antonio Campillo on the Occasion of his 65th Birthday. Springer, Cham, pp. 525-535. https://doi.org/10.1007/978-3-319-96827-8_23

The Poincaré polynomial of a linear code. / Galindo, Carlos; Hernando, Fernando ; Montserrat, Franciso et al.
Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics: Festschrift for Antonio Campillo on the Occasion of his 65th Birthday. ed. / Gert-Martin Greuel; Luis Narváez Macarro; Sebastià Xambó-Descamps. Cham: Springer, 2018. p. 525-535.

Research output: Chapter in Book/Report/Conference proceeding › Chapter › Academic › peer-review

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AU - Pellikaan, G.R.

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N2 - 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.

AB - 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.

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A2 - Narváez Macarro, Luis

A2 - Xambó-Descamps, Sebastià

PB - Springer

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The Poincaré polynomial of a linear code

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