galois.BCH.encode(message: ArrayLike, output: 'codeword' | 'parity' = 'codeword') → FieldArray

Encodes the message \(\mathbf{m}\) into the codeword \(\mathbf{c}\).

Shortened codes

For the shortened \([n-s,\ k-s,\ d]\) code (only applicable for systematic codes), pass \(k-s\) symbols into encode() to return the \(n-s\)-symbol message.

Parameters:
message: ArrayLike

The message as either a \(k\)-length vector or \((N, k)\) matrix, where \(N\) is the number of messages. For systematic codes, message lengths less than \(k\) may be provided to produce shortened codewords.

output: 'codeword' | 'parity' = 'codeword'

Specify whether to return the codeword or parity symbols only. The default is "codeword".

Returns:

If output="codeword", the codeword as either a \(n\)-length vector or \((N, n)\) matrix. If output="parity", the parity symbols as either a \(n-k\)-length vector or \((N, n-k)\) matrix.

Notes

The message vector \(\mathbf{m}\) is a member of \(\mathrm{GF}(q)^k\). The corresponding message polynomial \(m(x)\) is a degree-\(k\) polynomial over \(\mathrm{GF}(q)\).

\[\mathbf{m} = [m_{k-1},\ \dots,\ m_1,\ m_0] \in \mathrm{GF}(q)^k\]
\[m(x) = m_{k-1} x^{k-1} + \dots + m_1 x + m_0 \in \mathrm{GF}(q)[x]\]

The codeword vector \(\mathbf{c}\) is a member of \(\mathrm{GF}(q)^n\). The corresponding codeword polynomial \(c(x)\) is a degree-\(n\) polynomial over \(\mathrm{GF}(q)\).

\[\mathbf{c} = [c_{n-1},\ \dots,\ c_1,\ c_0] \in \mathrm{GF}(q)^n\]
\[c(x) = c_{n-1} x^{n-1} + \dots + c_1 x + c_0 \in \mathrm{GF}(q)[x]\]

The codeword vector is computed by matrix multiplication of the message vector with the generator matrix. The equivalent polynomial operation is multiplication of the message polynomial with the generator polynomial.

\[\mathbf{c} = \mathbf{m} \mathbf{G}\]
\[c(x) = m(x) g(x)\]

Examples

Encode a single message using the \(\textrm{BCH}(15, 7)\) code.

In [1]: bch = galois.BCH(15, 7)

In [2]: GF = bch.field

In [3]: m = GF.Random(bch.k); m
Out[3]: GF([1, 1, 0, 0, 0, 1, 0], order=2)

In [4]: c = bch.encode(m); c
Out[4]: GF([1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1], order=2)

Compute the parity symbols only.

In [5]: p = bch.encode(m, output="parity"); p
Out[5]: GF([1, 1, 1, 0, 1, 1, 1, 1], order=2)

Encode a single message using the shortened \(\textrm{BCH}(12, 4)\) code.

In [6]: bch = galois.BCH(15, 7)

In [7]: GF = bch.field

In [8]: m = GF.Random(bch.k - 3); m
Out[8]: GF([0, 1, 0, 1], order=2)

In [9]: c = bch.encode(m); c
Out[9]: GF([0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1], order=2)

Compute the parity symbols only.

In [10]: p = bch.encode(m, output="parity"); p
Out[10]: GF([0, 0, 1, 1, 0, 1, 1, 1], order=2)

Encode a matrix of three messages using the \(\textrm{BCH}(15, 7)\) code.

In [11]: bch = galois.BCH(15, 7)

In [12]: GF = bch.field

In [13]: m = GF.Random((3, bch.k)); m
Out[13]: 
GF([[0, 1, 1, 0, 0, 0, 0],
    [0, 1, 0, 1, 1, 0, 0],
    [1, 1, 1, 0, 1, 0, 0]], order=2)

In [14]: c = bch.encode(m); c
Out[14]: 
GF([[0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0],
    [0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1],
    [1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0]], order=2)

Compute the parity symbols only.

In [15]: p = bch.encode(m, output="parity"); p
Out[15]: 
GF([[0, 1, 0, 0, 1, 1, 1, 0],
    [1, 0, 0, 0, 1, 1, 1, 1],
    [0, 1, 0, 0, 0, 0, 0, 0]], order=2)

Encode a matrix of three messages using the shortened \(\textrm{BCH}(12, 4)\) code.

In [16]: bch = galois.BCH(15, 7)

In [17]: GF = bch.field

In [18]: m = GF.Random((3, bch.k - 3)); m
Out[18]: 
GF([[0, 0, 1, 0],
    [0, 0, 0, 0],
    [0, 0, 0, 0]], order=2)

In [19]: c = bch.encode(m); c
Out[19]: 
GF([[0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1],
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], order=2)

Compute the parity symbols only.

In [20]: p = bch.encode(m, output="parity"); p
Out[20]: 
GF([[0, 1, 1, 1, 0, 0, 1, 1],
    [0, 0, 0, 0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0, 0, 0, 0]], order=2)