Book contents
- Frontmatter
- Contents of Volume 1
- Contents of Volume 2
- Algebraic Geometry: A Celebration of Emma Previato’s 65th Birthday
- 1 Arithmetic Analogues of Hamiltonian Systems
- 2 Algebraic Spectral Curves over Q and their Tau-Functions
- 3 Frobenius Split Anticanonical Divisors
- 4 Halves of Points of an Odd Degree Hyperelliptic Curve in its Jacobian
- 5 Normal Forms for Kummer Surfaces
- 6 σ-Functions: Old and New Results
- 7 Bergman Tau-Function: From Einstein Equations and Dubrovin-Frobenius Manifolds to Geometry of Moduli Spaces
- 8 The Rigid Body Dynamics in an Ideal Fluid: Clebsch Top and Kummer Surfaces
- 9 An Extension of Delsarte, Goethals and Mac Williams Theorem on Minimal Weight Codewords to a Class of Reed-Muller Type Codes
- 10 A Primer on Lax Pairs
- 11 Lattice-Theoretic Characterizations of Classes of Groups
- 12 Jacobi Inversion Formulae for a Curve in Weierstrass Normal Form
- 13 Spectral Construction of Non-Holomorphic Eisenstein-Type Series and their Kronecker Limit Formula
- 14 Some Topological Applications of Theta Functions
- 15 Multiple Dedekind Zeta Values are Periods of Mixed Tate Motives
- 16 Noncommutative Cross-Ratio and Schwarz Derivative
9 - An Extension of Delsarte, Goethals and Mac Williams Theorem on Minimal Weight Codewords to a Class of Reed-Muller Type Codes
Published online by Cambridge University Press: 19 March 2020
- Frontmatter
- Contents of Volume 1
- Contents of Volume 2
- Algebraic Geometry: A Celebration of Emma Previato’s 65th Birthday
- 1 Arithmetic Analogues of Hamiltonian Systems
- 2 Algebraic Spectral Curves over Q and their Tau-Functions
- 3 Frobenius Split Anticanonical Divisors
- 4 Halves of Points of an Odd Degree Hyperelliptic Curve in its Jacobian
- 5 Normal Forms for Kummer Surfaces
- 6 σ-Functions: Old and New Results
- 7 Bergman Tau-Function: From Einstein Equations and Dubrovin-Frobenius Manifolds to Geometry of Moduli Spaces
- 8 The Rigid Body Dynamics in an Ideal Fluid: Clebsch Top and Kummer Surfaces
- 9 An Extension of Delsarte, Goethals and Mac Williams Theorem on Minimal Weight Codewords to a Class of Reed-Muller Type Codes
- 10 A Primer on Lax Pairs
- 11 Lattice-Theoretic Characterizations of Classes of Groups
- 12 Jacobi Inversion Formulae for a Curve in Weierstrass Normal Form
- 13 Spectral Construction of Non-Holomorphic Eisenstein-Type Series and their Kronecker Limit Formula
- 14 Some Topological Applications of Theta Functions
- 15 Multiple Dedekind Zeta Values are Periods of Mixed Tate Motives
- 16 Noncommutative Cross-Ratio and Schwarz Derivative
Summary
In 1970 Delsarte, Goethals and Mac Williams published a seminal paper on generalized Reed–Muller codes where, among many important results, they proved that the minimal weight codewords of these codes are obtained through the evaluation of certain polynomials which are a specific product of linear factors, which they describe. In the present paper we extend this result to a class of Reed–Muller type codes defined on a product of (possibly distinct) finite fields of the same characteristic. The paper also brings an expository section on the study of the structure of low weight codewords, not only for affine Reed–Muller type codes, but also for the projective ones.
- Type
- Chapter
- Information
- Integrable Systems and Algebraic Geometry , pp. 313 - 345Publisher: Cambridge University PressPrint publication year: 2020
- 1
- Cited by