Here you can fin the code in GAP that will give you the relations in the MZV. In order to use this code, download the files "MZVALLPROGRAM.G" and "LYNDONWORDS.TXT". After that open GAP and write:
gap> Read("MZVALLPROGRAM.G");
After that you will be able to generate the relations until degree "N" (define in "MZVALLPROGRAM.G") by doing:
gap> MzvRelationsWithRegularization(5, "TestDegree5.txt");
The output in the file TestDegree5.txt" looks like:
###############################################################
#-------------------------------------------------------
# Relations of degree 5:
#-------------------------------------------------------
with(LinearAlgebra):
sys := [
#-------------------------------------------------------
# Regularization
#-------------------------------------------------------
(-1)Z5+(1)Z4Z1+(1)Z3Z2+(1)Z2Z3=0,
(-1)Z4Z1+(-1)Z3Z2+(1)Z3Z1Z1+(1)Z2Z2Z1=0,
(-1)Z3Z2+(-1)Z2Z3+(1)Z2Z2Z1+(1)Z2Z1Z2=0,
(-1)Z3Z1Z1+(-1)Z2Z2Z1+(-1)Z2Z1Z2+(1)Z2Z1Z1Z1=0,
(-1)Z5+(6)Z4Z1+(2)Z3Z2=0,
(-1)Z4Z1+(6)Z3Z1Z1+(-1)Z2Z3+(1)Z2Z2Z1=0,
(-1)Z5+(6)Z4Z1+(2)Z3Z2=0,
(-1)Z4Z1+(6)Z3Z1Z1+(-1)Z2Z3+(1)Z2Z2Z1=0,
]:
var:=[Z5,Z4Z1,Z3Z2,Z3Z1Z1,Z2Z3,Z2Z2Z1,Z2Z1Z2,Z2Z1Z1Z1]:
A, b := GenerateMatrix(sys, var):
#A;
#ReducedRowEchelonForm(A);
ColumnDimension(A)-Rank(A);
###############################################################
This is the code for Maple, and ColumnDimension(A)-Rank(A) should be the dimension in Zagier's conjecture.
You also will be able to compute shuflle products as:
gap> shuffle(x*y, x*y);
(4)*x^2*y^2+(2)*(x*y)^2
gap> stuffle(x*y, x*y);
(1)*x^3*y+(2)*(x*y)^2
gap> shuffle(x*y, x*y)-stuffle(x*y,x*y);
(-1)*x^3*y+(4)*x^2*y^2