2024-02-14--ECC-01--ternary-code.ipynb

Kernel: Python 3 (ipykernel)

Let's create the field k having 3 elements, and the standard vector space V=k^9

K = GF(27);
V = VectorSpace(K,9)
print(K)
print(V)

Finite Field in z3 of size 3^3
Vector space of dimension 9 over Finite Field in z3 of size 3^3

Now let's create a certain 3 dimensional subspace C of V -- a code -- essentially by giving its generator matrix.

C= V.subspace([V([1,0,0,1,1,0,1,1,2]),
               V([0,1,0,1,0,1,1,2,1]),
               V([0,0,1,0,1,1,2,1,1])])
C

Vector space of degree 9 and dimension 3 over Finite Field in z3 of size 3^3
Basis matrix:
[1 0 0 1 1 0 1 1 2]
[0 1 0 1 0 1 1 2 1]
[0 0 1 0 1 1 2 1 1]

In order to manipulate the generator matrix as a matrix, we create the MatrixSpace of the right dimensions, and coerce the basis of C into a matrix:

MM = MatrixSpace(K,3,9)
G = MM.matrix(C.basis())
G

[1 0 0 1 1 0 1 1 2]
[0 1 0 1 0 1 1 2 1]
[0 0 1 0 1 1 2 1 1]

def weight(v):
    r = [x for x in v if x != 0]
    return len(r)

min([ weight(v) for v in C if v != 0])

6

Let's extract the matrix A which is the 3 x 6 matrix for which

G = [ I3 | A ]

A = MatrixSpace(K,3,6).matrix([b[3:9] for b in G])
A

[1 1 0 1 1 2]
[1 0 1 1 2 1]
[0 1 1 2 1 1]

And let's make the parity check matrix for C

i6=MatrixSpace(K,6,6).one()
H=block_matrix([[-A.transpose(),i6]],
               subdivide=False)        

H

[2 2 0 1 0 0 0 0 0]
[2 0 2 0 1 0 0 0 0]
[0 2 2 0 0 1 0 0 0]
[2 2 1 0 0 0 1 0 0]
[2 1 2 0 0 0 0 1 0]
[1 2 2 0 0 0 0 0 1]

And let's check that H * G.T == 0

H * G.T == 0

True

rank(H)

6

And indeed, if we use SAGE to check the right_kernel of the matrix H, we get exactly the subspace C.

H.right_kernel() == C