galois.BCH¶

class galois.BCH(n, k, primitive_poly=None, primitive_element=None, systematic=True)¶

A primitive, narrow-sense binary \(\textrm{BCH}(n, k)\) code.

A \(\textrm{BCH}(n, k)\) code is a \([n, k, d]_2\) linear block code with codeword size \(n\), message size \(k\), minimum distance \(d\), and symbols taken from an alphabet of size \(2\).

To create the shortened \(\textrm{BCH}(n-s, k-s)\) code, construct the full-sized \(\textrm{BCH}(n, k)\) code and then pass \(k-s\) bits into encode() and \(n-s\) bits into decode(). Shortened codes are only applicable for systematic codes.

Examples

Construct the BCH code.

In [1]: galois.bch_valid_codes(15)
Out[1]: [(15, 11, 1), (15, 7, 2), (15, 5, 3), (15, 1, 7)]

In [2]: bch = galois.BCH(15, 7); bch
Out[2]: <BCH Code: [15, 7, 5] over GF(2)>

Encode a message.

In [3]: m = galois.GF2.Random(bch.k); m
Out[3]: GF([0, 0, 0, 0, 1, 0, 0], order=2)

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

Corrupt the codeword and decode the message.

# Corrupt the first bit in the codeword
In [5]: c[0] ^= 1

In [6]: dec_m = bch.decode(c); dec_m
Out[6]: GF([0, 0, 0, 0, 1, 0, 0], order=2)

In [7]: np.array_equal(dec_m, m)
Out[7]: True

# Instruct the decoder to return the number of corrected bit errors
In [8]: dec_m, N = bch.decode(c, errors=True); dec_m, N
Out[8]: (GF([0, 0, 0, 0, 1, 0, 0], order=2), 1)

In [9]: np.array_equal(dec_m, m)
Out[9]: True

Constructors

`__init__`(n, k[, primitive_poly, ...])	Constructs a primitive, narrow-sense binary \(\textrm{BCH}(n, k)\) code.

Special Methods

`__repr__`()	Return repr(self).
`__str__`()	Return str(self).

Methods

`decode`()	Decodes the BCH codeword \(\mathbf{c}\) into the message \(\mathbf{m}\).
`detect`(codeword)	Detects if errors are present in the BCH codeword \(\mathbf{c}\).
`encode`(message[, parity_only])	Encodes the message \(\mathbf{m}\) into the BCH codeword \(\mathbf{c}\).

Attributes

`G`	The generator matrix \(\mathbf{G}\) with shape \((k, n)\).
`H`	The parity-check matrix \(\mathbf{H}\) with shape \((2t, n)\).
`d`	The design distance \(d\) of the \([n, k, d]_2\) code.
`field`	The Galois field array class for the \(\mathrm{GF}(2^m)\) field that defines the BCH code.
`generator_poly`	The generator polynomial \(g(x)\) whose roots are `roots`.
`is_narrow_sense`	Indicates if the BCH code is narrow sense, meaning the roots of the generator polynomial are consecutive powers of \(\alpha\) starting at 1, i.e. \(\alpha, \alpha^2, \dots, \alpha^{2t}\).
`is_primitive`	Indicates if the BCH code is primitive, meaning \(n = 2^m - 1\).
`k`	The message size \(k\) of the \([n, k, d]_2\) code
`n`	The codeword size \(n\) of the \([n, k, d]_2\) code
`roots`	The \(2t\) roots of the generator polynomial.
`systematic`	Indicates if the code is configured to return codewords in systematic form.
`t`	The error-correcting capability of the code.

__init__(n, k, primitive_poly=None, primitive_element=None, systematic=True)¶

Constructs a primitive, narrow-sense binary \(\textrm{BCH}(n, k)\) code.

Parameters

n : int: The codeword size \(n\), must be \(n = 2^m - 1\).
k : int: The message size \(k\).
primitive_poly : Optional[galois.Poly]: Optionally specify the primitive polynomial that defines the extension field \(\mathrm{GF}(2^m)\). The default is None which uses Matlab’s default, see galois.matlab_primitive_poly(). Matlab tends to use the lexicographically-minimal primitive polynomial as a default instead of the Conway polynomial.
primitive_element : Optional[Union[int, galois.Poly]]: Optionally specify the primitive element \(\alpha\) whose powers are roots of the generator polynomial \(g(x)\). The default is None which uses the lexicographically-minimal primitive element in \(\mathrm{GF}(2^m)\), see galois.primitive_element().
systematic : bool: Optionally specify if the encoding should be systematic, meaning the codeword is the message with parity appended. The default is True.

__repr__()¶: Return repr(self).

__str__()¶: Return str(self).

decode(codeword, errors=False)¶

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

Parameters

codeword : numpy.ndarray, galois.GF2: The codeword as either a \(n\)-length vector or \((N, n)\) matrix, where \(N\) is the number of codewords. For systematic codes, codeword lengths less than \(n\) may be provided for shortened codewords.
errors : bool, optional: Optionally specify whether to return the nubmer of corrected errors.

Returns

numpy.ndarray, galois.GF2 – The decoded message as either a \(k\)-length vector or \((N, k)\) matrix.
numpy.integer, numpy.ndarray – Optional return argument of the number of corrected bit errors as either a scalar or \(n\)-length vector. Valid number of corrections are in \([0, t]\). If a codeword has too many errors and cannot be corrected, -1 will be returned.

Notes

The codeword vector \(\mathbf{c}\) is defined as \(\mathbf{c} = [c_{n-1}, \dots, c_1, c_0] \in \mathrm{GF}(2)^n\), which corresponds to the codeword polynomial \(c(x) = c_{n-1} x^{n-1} + \dots + c_1 x + c_0\). The message vector \(\mathbf{m}\) is defined as \(\mathbf{m} = [m_{k-1}, \dots, m_1, m_0] \in \mathrm{GF}(2)^k\), which corresponds to the message polynomial \(m(x) = m_{k-1} x^{k-1} + \dots + m_1 x + m_0\).

In decoding, the syndrome vector \(s\) is computed by \(\mathbf{s} = \mathbf{c}\mathbf{H}^T\), where \(\mathbf{H}\) is the parity-check matrix. The equivalent polynomial operation is \(s(x) = c(x)\ \textrm{mod}\ g(x)\). A syndrome of zeros indicates the received codeword is a valid codeword and there are no errors. If the syndrome is non-zero, the decoder will find an error-locator polynomial \(\sigma(x)\) and the corresponding error locations and values.

For the shortened \(\textrm{BCH}(n-s, k-s)\) code (only applicable for systematic codes), pass \(n-s\) bits into decode() to return the \(k-s\)-bit message.

Examples

Vector

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

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

In [2]: GF = galois.GF(2)

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

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

Corrupt \(t\) bits of the codeword.

In [5]: bch.t
Out[5]: 2

In [6]: c[0:bch.t] ^= 1; c
Out[6]: GF([1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], order=2)

Decode the codeword and recover the message.

In [7]: d = bch.decode(c); d
Out[7]: GF([0, 0, 0, 0, 0, 0, 0], order=2)

In [8]: np.array_equal(d, m)
Out[8]: True

Decode the codeword, specifying the number of corrected errors, and recover the message.

In [9]: d, e = bch.decode(c, errors=True); d, e
Out[9]: (GF([0, 0, 0, 0, 0, 0, 0], order=2), 2)

In [10]: np.array_equal(d, m)
Out[10]: True

Vector (shortened)

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

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

In [12]: GF = galois.GF(2)

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

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

Corrupt \(t\) bits of the codeword.

In [15]: bch.t
Out[15]: 2

In [16]: c[0:bch.t] ^= 1; c
Out[16]: GF([1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1], order=2)

Decode the codeword and recover the message.

In [17]: d = bch.decode(c); d
Out[17]: GF([0, 1, 1, 0], order=2)

In [18]: np.array_equal(d, m)
Out[18]: True

Decode the codeword, specifying the number of corrected errors, and recover the message.

In [19]: d, e = bch.decode(c, errors=True); d, e
Out[19]: (GF([0, 1, 1, 0], order=2), 2)

In [20]: np.array_equal(d, m)
Out[20]: True

Matrix

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

In [21]: bch = galois.BCH(15, 7)

In [22]: GF = galois.GF(2)

In [23]: m = GF.Random((3, bch.k)); m
Out[23]: 
GF([[1, 0, 0, 1, 1, 0, 1],
    [0, 0, 0, 1, 0, 0, 1],
    [1, 0, 0, 0, 1, 1, 0]], order=2)

In [24]: c = bch.encode(m); c
Out[24]: 
GF([[1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0],
    [0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0],
    [1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1]], order=2)

Corrupt the codeword. Add zero errors to the first codeword, one to the second, and two to the third.

In [25]: c[1,0:1] ^= 1

In [26]: c[2,0:2] ^= 1

In [27]: c
Out[27]: 
GF([[1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0],
    [1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0],
    [0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1]], order=2)

Decode the codeword and recover the message.

In [28]: d = bch.decode(c); d
Out[28]: 
GF([[1, 0, 0, 1, 1, 0, 1],
    [0, 0, 0, 1, 0, 0, 1],
    [1, 0, 0, 0, 1, 1, 0]], order=2)

In [29]: np.array_equal(d, m)
Out[29]: True

Decode the codeword, specifying the number of corrected errors, and recover the message.

In [30]: d, e = bch.decode(c, errors=True); d, e
Out[30]: 
(GF([[1, 0, 0, 1, 1, 0, 1],
     [0, 0, 0, 1, 0, 0, 1],
     [1, 0, 0, 0, 1, 1, 0]], order=2),
 array([0, 1, 2]))

In [31]: np.array_equal(d, m)
Out[31]: True

Matrix (shortened)

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

In [32]: bch = galois.BCH(15, 7)

In [33]: GF = galois.GF(2)

In [34]: m = GF.Random((3, bch.k - 3)); m
Out[34]: 
GF([[1, 0, 0, 0],
    [1, 0, 1, 1],
    [1, 1, 1, 1]], order=2)

In [35]: c = bch.encode(m); c
Out[35]: 
GF([[1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1],
    [1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1],
    [1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1]], order=2)

Corrupt the codeword. Add zero errors to the first codeword, one to the second, and two to the third.

In [36]: c[1,0:1] ^= 1

In [37]: c[2,0:2] ^= 1

In [38]: c
Out[38]: 
GF([[1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1],
    [0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1],
    [0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1]], order=2)

Decode the codeword and recover the message.

In [39]: d = bch.decode(c); d
Out[39]: 
GF([[1, 0, 0, 0],
    [1, 0, 1, 1],
    [1, 1, 1, 1]], order=2)

In [40]: np.array_equal(d, m)
Out[40]: True

Decode the codeword, specifying the number of corrected errors, and recover the message.

In [41]: d, e = bch.decode(c, errors=True); d, e
Out[41]: 
(GF([[1, 0, 0, 0],
     [1, 0, 1, 1],
     [1, 1, 1, 1]], order=2),
 array([0, 1, 2]))

In [42]: np.array_equal(d, m)
Out[42]: True

detect(codeword)¶

Detects if errors are present in the BCH codeword \(\mathbf{c}\).

The \([n, k, d]_2\) BCH code has \(d_{min} \ge d\) minimum distance. It can detect up to \(d_{min}-1\) errors.

Parameters

codeword : Union[numpy.ndarray, galois.GF2]: The codeword as either a \(n\)-length vector or \((N, n)\) matrix, where \(N\) is the number of codewords. For systematic codes, codeword lengths less than \(n\) may be provided for shortened codewords.

Returns

A boolean scalar or array indicating if errors were detected in the corresponding codeword True or not False.

Return type

Union[numpy.bool_, numpy.ndarray]

Examples

Vector

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

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

In [2]: GF = galois.GF(2)

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

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

Detect no errors in the valid codeword.

In [5]: bch.detect(c)
Out[5]: False

Detect \(d_{min}-1\) errors in the codeword.

In [6]: bch.d
Out[6]: 5

In [7]: c[0:bch.d - 1] ^= 1; c
Out[7]: GF([1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0], order=2)

In [8]: bch.detect(c)
Out[8]: True

Vector (shortened)

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

In [9]: bch = galois.BCH(15, 7)

In [10]: GF = galois.GF(2)

In [11]: m = GF.Random(bch.k - 3); m
Out[11]: GF([1, 1, 1, 1], order=2)

In [12]: c = bch.encode(m); c
Out[12]: GF([1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1], order=2)

Detect no errors in the valid codeword.

In [13]: bch.detect(c)
Out[13]: False

Detect \(d_{min}-1\) errors in the codeword.

In [14]: bch.d
Out[14]: 5

In [15]: c[0:bch.d - 1] ^= 1; c
Out[15]: GF([0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1], order=2)

In [16]: bch.detect(c)
Out[16]: True

Matrix

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

In [17]: bch = galois.BCH(15, 7)

In [18]: GF = galois.GF(2)

In [19]: m = GF.Random((3, bch.k)); m
Out[19]: 
GF([[0, 1, 1, 1, 0, 1, 0],
    [0, 0, 0, 1, 0, 0, 0],
    [1, 1, 1, 1, 1, 0, 0]], order=2)

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

Detect no errors in the valid codewords.

In [21]: bch.detect(c)
Out[21]: array([False, False, False])

Detect one, two, and \(d_{min}-1\) errors in the codewords.

In [22]: bch.d
Out[22]: 5

In [23]: c[0,0:1] ^= 1

In [24]: c[1,0:2] ^= 1

In [25]: c[2, 0:bch.d - 1] ^= 1

In [26]: c
Out[26]: 
GF([[1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0],
    [1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1],
    [0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1]], order=2)

In [27]: bch.detect(c)
Out[27]: array([ True,  True,  True])

Matrix (shortened)

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

In [28]: bch = galois.BCH(15, 7)

In [29]: GF = galois.GF(2)

In [30]: m = GF.Random((3, bch.k - 3)); m
Out[30]: 
GF([[1, 0, 0, 0],
    [1, 1, 1, 0],
    [0, 0, 0, 1]], order=2)

In [31]: c = bch.encode(m); c
Out[31]: 
GF([[1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1],
    [1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0],
    [0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1]], order=2)

Detect no errors in the valid codewords.

In [32]: bch.detect(c)
Out[32]: array([False, False, False])

Detect one, two, and \(d_{min}-1\) errors in the codewords.

In [33]: bch.d
Out[33]: 5

In [34]: c[0,0:1] ^= 1

In [35]: c[1,0:2] ^= 1

In [36]: c[2, 0:bch.d - 1] ^= 1

In [37]: c
Out[37]: 
GF([[0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1],
    [0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0],
    [1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1]], order=2)

In [38]: bch.detect(c)
Out[38]: array([ True,  True,  True])

encode(message, parity_only=False)¶

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

Parameters

message : Union[numpy.ndarray, galois.GF2]: 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.
parity_only : bool: Optionally specify whether to return only the parity bits. This only applies to systematic codes. The default is False.

Returns

The codeword as either a \(n\)-length vector or \((N, n)\) matrix. The return type matches the message type. If parity_only=True, the parity bits are returned as either a \(n - k\)-length vector or \((N, n-k)\) matrix.

Return type

Union[numpy.ndarray, galois.GF2]

Notes

The message vector \(\mathbf{m}\) is defined as \(\mathbf{m} = [m_{k-1}, \dots, m_1, m_0] \in \mathrm{GF}(2)^k\), which corresponds to the message polynomial \(m(x) = m_{k-1} x^{k-1} + \dots + m_1 x + m_0\). The codeword vector \(\mathbf{c}\) is defined as \(\mathbf{c} = [c_{n-1}, \dots, c_1, c_0] \in \mathrm{GF}(2)^n\), which corresponds to the codeword polynomial \(c(x) = c_{n-1} x^{n-1} + \dots + c_1 x + c_0\).

The codeword vector is computed from the message vector by \(\mathbf{c} = \mathbf{m}\mathbf{G}\), where \(\mathbf{G}\) is the generator matrix. The equivalent polynomial operation is \(c(x) = m(x)g(x)\). For systematic codes, \(\mathbf{G} = [\mathbf{I}\ |\ \mathbf{P}]\) such that \(\mathbf{c} = [\mathbf{m}\ |\ \mathbf{p}]\). And in polynomial form, \(p(x) = -(m(x) x^{n-k}\ \textrm{mod}\ g(x))\) with \(c(x) = m(x)x^{n-k} + p(x)\). For systematic and non-systematic codes, each codeword is a multiple of the generator polynomial, i.e. \(g(x)\ |\ c(x)\).

For the shortened \(\textrm{BCH}(n-s, k-s)\) code (only applicable for systematic codes), pass \(k-s\) bits into encode() to return the \(n-s\)-bit codeword.

Examples

Vector

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

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

In [2]: GF = galois.GF(2)

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

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

Compute the parity bits only.

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

Vector (shortened)

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

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

In [7]: GF = galois.GF(2)

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

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

Compute the parity bits only.

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

Matrix

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

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

In [12]: GF = galois.GF(2)

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

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

Compute the parity bits only.

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

Matrix (shortened)

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

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

In [17]: GF = galois.GF(2)

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

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

Compute the parity bits only.

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

property G : galois.GF2¶

The generator matrix \(\mathbf{G}\) with shape \((k, n)\).

Examples

In [1]: bch = galois.BCH(15, 7); bch
Out[1]: <BCH Code: [15, 7, 5] over GF(2)>

In [2]: bch.G
Out[2]: 
GF([[1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0],
    [0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0],
    [0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0],
    [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1],
    [0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0],
    [0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1],
    [0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1]], order=2)

property H : galois.FieldArray¶

The parity-check matrix \(\mathbf{H}\) with shape \((2t, n)\).

Examples

In [1]: bch = galois.BCH(15, 7); bch
Out[1]: <BCH Code: [15, 7, 5] over GF(2)>

In [2]: bch.H
Out[2]: 
GF([[ 9, 13, 15, 14,  7, 10,  5, 11, 12,  6,  3,  8,  4,  2,  1],
    [13, 14, 10, 11,  6,  8,  2,  9, 15,  7,  5, 12,  3,  4,  1],
    [15, 10, 12,  8,  1, 15, 10, 12,  8,  1, 15, 10, 12,  8,  1],
    [14, 11,  8,  9,  7, 12,  4, 13, 10,  6,  2, 15,  5,  3,  1]],
   order=2^4)

property d : int¶

The design distance \(d\) of the \([n, k, d]_2\) code. The minimum distance of a BCH code may be greater than the design distance, \(d_{min} \ge d\).

Examples

In [1]: bch = galois.BCH(15, 7); bch
Out[1]: <BCH Code: [15, 7, 5] over GF(2)>

In [2]: bch.d
Out[2]: 5

property field : galois.FieldClass¶

The Galois field array class for the \(\mathrm{GF}(2^m)\) field that defines the BCH code.

Examples

In [1]: bch = galois.BCH(15, 7); bch
Out[1]: <BCH Code: [15, 7, 5] over GF(2)>

In [2]: bch.field
Out[2]: <class 'numpy.ndarray over GF(2^4)'>

In [3]: print(bch.field)
Galois Field:
  name: GF(2^4)
  characteristic: 2
  degree: 4
  order: 16
  irreducible_poly: x^4 + x + 1
  is_primitive_poly: True
  primitive_element: x

property generator_poly : galois.Poly¶

The generator polynomial \(g(x)\) whose roots are roots.

Examples

In [1]: bch = galois.BCH(15, 7); bch
Out[1]: <BCH Code: [15, 7, 5] over GF(2)>

In [2]: bch.generator_poly
Out[2]: Poly(x^8 + x^7 + x^6 + x^4 + 1, GF(2))

# Evaluate the generator polynomial at its roots in GF(2^m)
In [3]: bch.generator_poly(bch.roots, field=bch.field)
Out[3]: GF([0, 0, 0, 0], order=2^4)

property is_narrow_sense : bool¶

Indicates if the BCH code is narrow sense, meaning the roots of the generator polynomial are consecutive powers of \(\alpha\) starting at 1, i.e. \(\alpha, \alpha^2, \dots, \alpha^{2t}\).

Examples

In [1]: bch = galois.BCH(15, 7); bch
Out[1]: <BCH Code: [15, 7, 5] over GF(2)>

In [2]: bch.is_narrow_sense
Out[2]: True

In [3]: bch.roots
Out[3]: GF([2, 4, 8, 3], order=2^4)

In [4]: bch.field.primitive_element**(np.arange(1, 2*bch.t + 1))
Out[4]: GF([2, 4, 8, 3], order=2^4)

property is_primitive : bool¶

Indicates if the BCH code is primitive, meaning \(n = 2^m - 1\).

Examples

In [1]: bch = galois.BCH(15, 7); bch
Out[1]: <BCH Code: [15, 7, 5] over GF(2)>

In [2]: bch.is_primitive
Out[2]: True

property k : int¶

The message size \(k\) of the \([n, k, d]_2\) code

Examples

In [1]: bch = galois.BCH(15, 7); bch
Out[1]: <BCH Code: [15, 7, 5] over GF(2)>

In [2]: bch.k
Out[2]: 7

property n : int¶

The codeword size \(n\) of the \([n, k, d]_2\) code

Examples

In [1]: bch = galois.BCH(15, 7); bch
Out[1]: <BCH Code: [15, 7, 5] over GF(2)>

In [2]: bch.n
Out[2]: 15

property roots : galois.FieldArray¶

The \(2t\) roots of the generator polynomial. These are consecutive powers of \(\alpha\), specifically \(\alpha, \alpha^2, \dots, \alpha^{2t}\).

Examples

In [1]: bch = galois.BCH(15, 7); bch
Out[1]: <BCH Code: [15, 7, 5] over GF(2)>

In [2]: bch.roots
Out[2]: GF([2, 4, 8, 3], order=2^4)

# Evaluate the generator polynomial at its roots in GF(2^m)
In [3]: bch.generator_poly(bch.roots, field=bch.field)
Out[3]: GF([0, 0, 0, 0], order=2^4)

property systematic : bool¶

Indicates if the code is configured to return codewords in systematic form.

Examples

In [1]: bch = galois.BCH(15, 7); bch
Out[1]: <BCH Code: [15, 7, 5] over GF(2)>

In [2]: bch.systematic
Out[2]: True

property t : int¶

The error-correcting capability of the code. The code can correct \(t\) bit errors in a codeword.

Examples

In [1]: bch = galois.BCH(15, 7); bch
Out[1]: <BCH Code: [15, 7, 5] over GF(2)>

In [2]: bch.t
Out[2]: 2

Last update: Apr 03, 2022