Coding theory package for Macaulay2

  title={Coding theory package for Macaulay2},
  author={Taylor Ball and Eduardo Camps and Henry Chimal-Dzul and Delio Jaramillo-Velez and Hiram H. L'opez and Nathan S. Nichols and Matthew Perkins and Ivan Soprunov and German Vera-Mart'inez and Gwyneth R. Whieldon},
In this Macaulay2 \cite{M2} package we define an object called {\it linear code}. We implement functions that compute basic parameters and objects associated with a linear code, such as generator and parity check matrices, the dual code, length, dimension, and minimum distance, among others. We define an object {\it evaluation code}, a construction which allows to study linear codes using tools of algebraic geometry and commutative algebra. We implement functions to generate important families… 
Multivariate Goppa codes
The multivariate Goppa codes are utilized to obtain entanglement-assisted quantum error-correcting codes and to build families of long LCD, self-dual, or self-orthogonal codes.
The dual of an evaluation code
This work shows that the dual of an evaluation code is the evaluation code of the algebraic dual, and provides an explicit duality criterion in terms of the vnumber and the Hilbert function of a vanishing ideal for Reed–Muller-type codes corresponding to Gorenstein ideals.
Relative generalized Hamming weights of evaluation codes
The aim of this work is to algebraically describe the relative generalized Hamming weights of evaluation codes in terms of a footprint bound and it is proved that this bound can be sharp.


The Theory of Error-Correcting Codes
Toric Surface Codes and Minkowski Sums
Upper and lower bounds on the minimum distance of a toric code constructed from a polygon $P \subset {\mathbb R}^2$ are obtained by examining Minkowski sum decompositions of subpolygons of $P$.
A family of optimal locally recoverable codes
  • Itzhak Tamo, A. Barg
  • Computer Science
    2014 IEEE International Symposium on Information Theory
  • 2014
A family of LRC codes that attain the maximum possible value of the distance for a given locality parameter and code cardinality are presented.
Projective Nested Cartesian Codes
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.
Minimum distance functions of graded ideals and Reed-Muller-type codes
On the parameters of r-dimensional toric codes
  • D. Ruano
  • Mathematics
    Finite Fields Their Appl.
  • 2007
Fundamentals of Error-Correcting Codes
This chapter discusses the development of soft decision and iterative decoding in linear codes, as well as some of the techniques used in convolutional codes.
Algebraic methods for parameterized codes and invariants of vanishing ideals over finite fields
Toric Surfaces and Error-correcting Codes
From an integral convex polytope in ℝ2 we give an explicit description of an error-correcting code over the finite field \( {{\Bbb F}_q} \) of length (q — 1)2. The codes are obtained from toric