galois.BCH.encode(output: Literal[codeword] | Literal[parity] = 'codeword')

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

Parameters:
message: ArrayLike

The message as either a $$k$$-length vector or $$(N, k)$$ matrix, where $$N$$ is the number of messages.

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.

output: Literal[codeword] | Literal[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, 0, 1, 0, 1, 0, 0], order=2)

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


Compute the parity symbols only.

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

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


Compute the parity symbols only.

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

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


Compute the parity symbols only.

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

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


Compute the parity symbols only.

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