The idea

We want to transmit information over a potentially noisy channel. So we want to encode our information in some way that permits us to later decode it even in the presence of transmission errors.

We want to exploit algebra to create our codes. We will use as our alphabet a finite field \(K = \mathbb{F}_q\)

Some recollections about finite fields

We pause to recall some information about finite fields. We plan to return to this topic.

First of all, any finite field \(K\) contains a copy of the field \(\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}\) for some prime number \(p>0\). Moreover, \(K\) can then be viewed as a finite-dimensional vector space over \(\mathbb{F}_p\); as such, if \(d = \dim_{\mathbb{F}_p} K\) we see that \(\#K = |K| = p^d\).

Thus every finite field has order a power of some prime number \(p\).

On the other hand, for any prime power \(q = p^d\) there is a finite field \(\mathbb{F}_q\) which is unique up to isomorphism.

For example, \(\mathbb{F}_9 = \mathbb{F}_3[i] = \mathbb{F}_3 + \mathbb{F}_3 i\) where \(i^2 = -1 = 2 \in \mathbb{F}_3\). In fact, \[\mathbb{F}_9 \simeq \mathbb{F}_3[T]/\langle T^2 + 1\rangle\] (quotient of the polynomial ring \(\mathbb{F}_3[T]\) by the principal ideal generated by the minimal polynomial \(T^2 + 1\) of the element \(i \in \mathbb{F}_9\). This isomorphism identifies \(i\) with the coset \(T + \langle T^2 + 1 \rangle\)).

Codewords as vectors

We are going to study what are known as linear codes \(C\). A linear code \(C\) is a linear subspace \(C \subseteq \mathbb{F}_q^n\) for some natural number \(n\).

Thus a code word is a vector \(\mathbf{v} \in C\), and we can write \(\mathbf{v} = (v_1, v_2, \cdots, v_n)\) for elements \(v_i\) in our alphabet \(k = \mathbb{F}_q\).

We write \(k\) for the dimension of \(C\); i.e. \(k = \dim_{\mathbb{F}_q} C\). We say that \(C\) is an \([n,k]\)-code, or more precisely an \([n,k]_q\)-code.

Specifying a code via a generator matrix.

Let \(C\) be an \([n,k]_q\)-code, and choose a basis \(b_1,\cdots,b_k\) for \(C\) as \(\mathbb{F}_q\)-vector space.

Since \(C \subset \mathbb{F}_q^n = \mathbb{F}_q^{1 \times n}\), we view elements of \(C\) as \(1 \times n\) row vectors.

Now form the matrix \(k \times n\) matrix \(G\) whose rows are the \(1 \times n\) vectors \(\mathbf{b}_1,\mathbf{b}_2,\cdots,\mathbf{b}_k\):

\[G = \begin{bmatrix} \mathbf{b}_1 \\ \mathbf{b}_2 \\ \vdots \\ \mathbf{b}_k \end{bmatrix}.\]

\(G\) is known as a generator matrix for the code \(C\).

Notice that we recover \(C\) from \(G\) as the image of the linear transformation \(\mathbb{F}_q^k \to \mathbb{F}_q^n\) determined by right-multiplication with \(G\):

\[C = \{ \mathbf{x} G \vert \mathbf{x} \in \mathbb{F}_q^{1 \times n}\} = \operatorname{image}(\mathbf{x} \mapsto \mathbf{x}G).\]

Standard form

Let us write \(\mathbf{e}_1,\mathbf{e}_2,\cdots,\mathbf{e}_n\) for the standard basis for \(\mathbb{F}_q^n\).

Lemma:

Let \(C\) be an \([n,k]_q\)-code, and let \(G\) be a generator matrix for \(C\).

1. After possibly replacing the basis \(\mathbf{e}_1,\cdots,\mathbf{e}_n\) with the basis \(\mathbf{e}_{\sigma(1)},\cdots,\mathbf{e}_{\sigma(n)}\) for some \(\sigma \in S_n\), we may suppose that the first \(k\)-columns of the \(k\times n\) matrix are linearly independent.

2. If the conclusion a. holds, then \(C\) has a generator matrix of the form \[G' = \left [ \begin{array}{l|l} \mathbf{I}_k & A \end{array} \right ]\] for some \(k \times (n-k)\) matrix \(A\), where \(\mathbf{I}_k\) denotes the \(k \times k\) identity matrix.

Proof of the Lemma:

1. By construction, \(G\) has \(k\) linearly independent rows (its rows are a basis for \(C\)). Since \(G\) is \(k \times n\) it follows that the rank of \(G\) is equal to \(k\).

   Since the row rank of a matrix is equal to the column rank of the matrix, it follows that \(G\) has \(k\) linearly independent columns. After possibly re-ordering the order of these columns, we may arrange that \(G\) has the required form.

2. Suppose that the first \(k\)-columns of \(G\) are linearly independent and consider the projection mapping \[\pi:\mathbb{F}_q^n \to \mathbb{F}_q^k \quad \text{given by the rule} \quad \pi(x_1,\cdots,x_n) = (x_1, \cdots,x_k).\]

   Write \(\b_1,\b_2,\cdots,b_k \in \mathbb{F}_q^{1 \times n}\) for the rows of \(G\). Since the first \(k\)-columns of \(G\) are linearly independent, the rank of the \(k\times k\) matrix whose rows are the \(1 \times k\)-vectors \(\pi(\b_1), \pi(\b_2), \cdots, \pi(\b_k)\) is equal to \(k\).

   This shows that the dimension of \(\pi(C)\) is \(\ge k\), and hence that \(\pi(C) = \mathbb{F}_q^k\). In other words, the restriction \(\pi_{\mid C}:C \to \mathbb{F}_q^k\) of \(\pi\) to \(C\) is surjective.

   In view of the surjectivity, for \(i = 1,2,\cdots,k\) we may choose a vector \(\b_i' \in \pi^{-1}(\mathbf{e}_i) \cap C\).

   Let \(G'\) whose rows are the vectors \(\b_i'\): \[G' = \begin{bmatrix} \b_1' \\ \b_2' \\ \vdots \\ \b_k' \end{bmatrix}.\] Since \[\begin{bmatrix} \pi(\b_1') \\ \pi(\b_2') \\ \vdots \\ \pi(\b_k') \end{bmatrix} = \mathbf{I}_k,\] \(G'\) has the required properties.

We say that a (\(k \times n\)) generator matrix \(G\) for the \([n,k]_q\)-code \(C\) is in standard form if it has the form \[G = \left [ \begin{array}{l|l} \mathbf{I}_k & A \end{array} \right ]\] for some \(k \times (n-k)\) matrix \(A\); thus the Lemma asserts that (after possibly changing the ordering of the coordinates on \(\mathbb{F}_q^n\)) every code \(C\) has a generator matrix in standard form.

Check matrices

The preceding discussion described the subspace \(C\) by giving generators for the vector space. In contrast, we may also specify a subspace using linear equations.

So: let \(C\) be an \([n,k]_q\)-code.

Consider the quotient linear mapping \(x \mapsto x + C\): \[\mathbb{F}_q^n \to \mathbb{F}_q^n/C \simeq \mathbb{F}_q^{n-k}\]

There is an \((n-k) \times n\) matrix \(H\) which represents this linear mapping; then \[C = \operatorname{Null}(H) = \{x \in \mathbb{F}_q^{1 \times n} \mid Hx^T = 0\};\] i.e. we see that \(C\) is the null space for some \((n-k) \times n\) matrix \(H\).

We say that such a matrix is a check matrix for \(C\). For a vector \(x \in \mathbb{F}_q^{1 \times n}\), we can check membership in \(C\) using \(H\): we have \(x \in C \iff Hx^T = 0\).

Proposition:

Suppose that \(C\) is an \([n,k]_q\)-code and that \[G = \left [ \begin{array}{l|l} \mathbf{I}_k & A \end{array} \right ]\] is a generator matrix for \(C\) in standard form. Then the \((n-k) \times n\) matrix \[H = \left [ \begin{array}{l|l} -A^T & \mathbf{I}_{n-k} \end{array} \right ]\] is a check matrix for \(C\).

Proof:

We observe that \[H \cdot G^T = \left [ \begin{array}{l|l} -A^T & \mathbf{I}_{n-k} \end{array} \right ] \left [ \begin{array}{l|l} \mathbf{I}_k \\ \hline A^T \end{array} \right ] = -A^T \cdot \mathbf{I}_k + \mathbf{I}_{n-k} \cdot A^T = -A^T + A^T = \mathbf{0}.\]

Since the rows of \(G\) are a basis for \(C\), this shows that \(Hx^T = 0\) for every \(x \in C\), i.e. \(C \subset \operatorname{Null}(H)\).

Now, \(H\) clearly has rank \((n-k)\), so \(\dim C = \dim \operatorname{Null}(H)\) and hence \(C = \operatorname{Null}(H)\). This completes the proof.

Weights and distance

For a vector \(v = (v_1,v_2,\cdots,v_n) \in \mathbb{F}_q^n = \mathbb{F}_q^{1 \times n}\) we define \(\operatorname{weight}(v)\) to be the number of non-zero entries; i.e. \[\operatorname{weight}(v) = \# \{ i \mid v_i \ne 0\}.\]

For two vectors \(v,w \in \mathbb{F}_q^n\) the distance between \(v\) and \(w\) is defined to be \[\operatorname{dist}(v,w) = \operatorname{weight}(v-w).\]

Thus \(\operatorname{dist}(v,w)\) represents the number of coordinates in which \(v\) and \(w\) differ.

For a subspace \(C \subset \mathbb{F}_q^n\) - i.e. a code - we define the minimal distance of \(C\) to be \[d = \min \{ \operatorname{dist}(v,w) \mid v,w \in C, v \ne w\}.\]

The following lemma is immediate:

Lemma:

\(d = \min \{ \operatorname{weight}(v) \mid v \in C, v \ne 0\}.\)

If \(d\) is the minimal distance of the \([n,k]_q\) code \(C\), we say that \(C\) is an \([n,k,d]_q\)-code.

Example

We investigate an example using SageMath.

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 \([9,3]_3\) 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]

We investigate the weights of non-zero vectors in C:

# count the non-zero entries in a vector
def weight(v):
    r = [x for x in v if x != 0]
    return len(r)

# we now find the minimum of the weight of v for non-zero vectors v in C
min([ weight(v) for v in C if v != 0 ])
=> 6

This shows that \(C\) is a \([9,3,6]_3\)-code.

Notice that the generator matrix G is in standard form. Let’s extract from G 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]

We can now form the 6 × 9 check matrix H = [ -A.T | I₆] as above.

# form the 6x6 identity matrix
i6=MatrixSpace(K,6,6).one()

# we construct H via *block_matrix*
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]

We can confirm that H * G.T == 0 and that H has rank 6:

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
=>
True

So H is indeed a check-matrix for the code C.

Bibliography