EvaluationCode -- class of evaluation codes

Description

EvaluationCode is the class of linear codes obtained by evaluating polynomials in $F[X_1,\ldots,X_m]$, where $F$ is a finite field, at a set of points in $F^m$. There are various constructions of evaluation codes depending on how the polynomials and points are chosen. Important examples include Reed-Solomon codes, Reed-Muller codes, monomial codes, Cartesian codes, and toric codes. To construct a linear code, see evaluationCode.

The basic structure is a hash table. One of the values is the resulting linear code of type LinearCode. Other values include the set of points, its vanishing ideal, the set of polynomials, and more.

i1 : F=GF(4);

i2 : R=F[x,y];

i3 : P={{0,0},{1,0},{0,1},{a,a}};

i4 : S={x+y,x^2+y^2, a+x*y^2};

i5 : C=evaluationCode(F,P,S);

i6 : C.VanishingIdeal

                          2              2    2
o6 = ideal (x*y + (a + 1)y  + (a + 1)y, x  + y  + x + y)

o6 : Ideal of R

i7 : C.PolynomialSet

              2    2     2
o7 = {x + y, x  + y , x*y  + a}

o7 : List

i8 : C.LinearCode

                                  4
o8 = LinearCode{AmbientModule => F                                          }
                BaseField => F
                cache => CacheTable{}
                Code => image | 0 0 a   |
                              | 1 1 a   |
                              | 1 1 a   |
                              | 0 0 a+1 |
                GeneratorMatrix => | 0 1 1 0   |
                                   | 0 1 1 0   |
                                   | a a a a+1 |
                Generators => {{0, 1, 1, 0}, {0, 1, 1, 0}, {a, a, a, a + 1}}
                ParityCheckMatrix => | 1 0 0 a+1 |
                                     | 0 1 1 0   |
                ParityCheckRows => {{1, 0, 0, a + 1}, {0, 1, 1, 0}}

o8 : LinearCode

i9 : length C.LinearCode

o9 = 4

i10 : dim C.LinearCode

o10 = 2

Functions and methods returning an object of class EvaluationCode :

cartesianCode(Ring,List,List) -- see cartesianCode -- Cartesian code
cartesianCode(Ring,List,Matrix) -- see cartesianCode -- Cartesian code
cartesianCode(Ring,List,ZZ) -- see cartesianCode -- Cartesian code
evaluationCode(Ring,List,List) -- see evaluationCode -- functions to construct evaluation codes over Galois fields
evaluationCode(Ring,List,Matrix) -- see evaluationCode -- functions to construct evaluation codes over Galois fields
orderCode(Ideal,List,List,ZZ) -- see orderCode -- computes an order code for a given weight
orderCode(Ideal,List,ZZ) -- see orderCode -- computes an order code for a given weight
orderCode(Ring,List,List,ZZ) -- see orderCode -- computes an order code for a given weight
reedMullerCode -- constructs the Reed-Muller code
reedMullerCode(ZZ,ZZ,ZZ) -- see reedMullerCode -- constructs the Reed-Muller code
reedSolomonCode -- constructs the Reed-Solomon code
reedSolomonCode(Ring,List,ZZ) -- see reedSolomonCode -- constructs the Reed-Solomon code
toricCode(Ring,Matrix) -- see toricCode -- a toric code construction

For the programmer

The object EvaluationCode is a type, with ancestor classes HashTable < Thing.

evaluationCode -- functions to construct evaluation codes over Galois fields
footPrint -- generalized footprint function of an ideal
genMinDisIdeal -- generalized minimum distance function of an ideal
hyp -- hyp function of an ideal
vasconcelosDegree -- Vasconcelos function of an ideal
vNumber -- the v-number of a graded ideal

Symbols that are used as a key for storing information of an evaluation code

PolynomialSet -- a set of polynomials for an evaluation code
VanishingIdeal -- vanishing ideal of an evaluation code

EvaluationCode -- class of evaluation codes

Description

Functions and methods returning an object of class EvaluationCode :

For the programmer

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Functions related to ideals and evaluation codes

Symbols that are used as a key for storing information of an evaluation code