# Decoding Affine Variety Codes Using Gröbner Bases

@article{Fitzgerald1998DecodingAV,
title={Decoding Affine Variety Codes Using Gr{\"o}bner Bases},
author={J. Fitzgerald and Robert F. Lax},
journal={Designs, Codes and Cryptography},
year={1998},
volume={13},
pages={147-158}
}
• Published 1 February 1998
• Computer Science
• Designs, Codes and Cryptography
We define a class of codes that we call affine variety codes. These codes are obtained by evaluating functions in the coordinate ring of an affine variety on all the Fq-rational points of the variety. We show that one can, at least in theory, decode these codes up to half the true minimum distance by using the theory of Gröbner bases. We extend results of A. B. Cooper and of X. Chen, I. S. Reed, T. Helleseth, and T. K. Truong.
72 Citations

### Bounding the Minimum Distance of Affine Variety Codes Using Symbolic Computations of Footprints

• Computer Science
ICMCTA
• 2017
A new method inspired by Buchbergers algorithm is developed where the codes constructed have parameters as good as the best known codes according to Grobner basis theory and in the remaining few cases the parameters are almost as good.

### On the weights of affine-variety codes and some Hermitian codes

• Computer Science
• 2011
For any affine-variety code it is shown how to construct an ideal whose solutions correspond to codewords with any assigned weight, and the number of minimum-weight code- words for all Hermitian codes with d ≤ q is determined.

### Affine Variety Codes over a Hyperelliptic Curve

• Computer Science, Mathematics
Probl. Inf. Transm.
• 2021
The proposed method to calculate the generalized Hamming weights can be applied to any primary monomial affine variety codes and obtain lower bounds on the generalizedHamming weights of the constructed codes.

### Projective Nested Cartesian Codes

• Computer Science
• 2017
An upper bound for the minimum distance and the exact minimum distance in a special case (which includes the projective Reed–Muller codes) is calculated and some relations between the parameters of these codes and those of the affine cartesian codes are shown.

### On affine variety codes from the Klein quartic

• Computer Science
Cryptography and Communications
• 2018
A new method is developed where the code parameters are inspired by Buchberger's algorithm and almost all have parameters as good as the best known codes according to Grassl (2007) and in the remaining few cases the parameters are almost as good.

### An Examination of Quaternary Higher-Dimensional Affine Variety Codes with an Improved Minimum Distance Bound

• Computer Science
• 2002
This paper examines several families of affine variety codes [6] in which the minimum distance is optimal given fixed parameters of length and dimension. We also introduce a bound on the minimum

### On the Hermitian curve, its intersections with some conics and their applications to affine-variety codes and Hermitian codes

• Computer Science, Mathematics
• 2012
The number of minimum-weight codewords for all Hermitian codes with $d\leq q$ and all second- Weight codeword for distance-$3,4$ codes are determined.

### Toric Codes over Finite Fields

• D. Joyner
• Computer Science
Applicable Algebra in Engineering, Communication and Computing
• 2004
Abstract.In this note, a class of error-correcting codes is associated to a toric variety defined over a finite field q, analogous to the class of AG codes associated to a curve. For small q, many of

## References

SHOWING 1-10 OF 31 REFERENCES

### Which linear codes are algebraic-geometric?

• Computer Science, Mathematics
IEEE Trans. Inf. Theory
• 1991
An infinite series of curves is constructed in order to show that all linear codes can be obtained from curves using Goppa's construction, and it is proven that this triple is in a certain sense unique in the case of the (7,4,3) code.

### Improved geometric Goppa codes. I. Basic theory

• Computer Science, Mathematics
IEEE Trans. Inf. Theory
• 1995
For these improved geometric Goppa codes, a designed minimum distance can be easily determined and a decoding procedure which corrects up to half the designed minimum Distance is also given.

### Algebraic-Geometry Codes

• Computer Science
IEEE Trans. Inf. Theory
• 1998
Elements of the theory of algebraic curves, at a level sufficient to understand the code constructions and decoding algorithms, are introduced.

### On the Decoding of Cyclic Codes Using Gröbner Bases

• Computer Science
Applicable Algebra in Engineering, Communication and Computing
• 1997
An algorithm for decoding cyclic codes up to their true minimum distance using Gröbner basis techniques is revisited and a geometric characterization of the number of errors is given, and the corresponding algebraic characterization is analyzed.

### Use of Grobner bases to decode binary cyclic codes up to the true minimum distance

• Computer Science
IEEE Trans. Inf. Theory
• 1994
It is proved here that the process of transforming F to the normalized reduced Grobner basis of I(F) with respect to the "purely lexicographical" ordering automatically converges to L(z).

### Algebraic-geometric codes and multidimensional cyclic codes: a unified theory and algorithms for decoding using Grobner bases

• Computer Science
IEEE Trans. Inf. Theory
• 1995
It is proved that any algebraic-geometric (AG) code can be expressed as a cross section of an extended multidimensional cyclic code, and Sakata's algorithm can be used to find linear recursion relations which hold on the syndrome array.

### Toward a New Method of Decoding Algebraic Codes Using Groebner Bases

A binary BCH error control code is a vector subspace of binary N- tuples that is decoded by computing a set of syndrome equations which are multivariate polynomials over GF(2m) and which exhibit a certain symmetry.

### General principles for the algebraic decoding of cyclic codes

• Computer Science
IEEE Trans. Inf. Theory
• 1994
It is shown that from a polynomial ideal point of view, the decoding problems of cyclic codes are closely related to the monic generators of certain polynometric ideals.

### Introduction to coding theory and algebraic geometry

• Computer Science
• 1989
This chapter discusses coding theory, algebraic geometry, and Shimura curves and codes, as well as some examples of coding in the real world.