galois.BCH

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

Constructs 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.

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.

Parameters

• n (int) – The codeword size \(n\), must be \(n = 2^m - 1\).

• k (int) – The message size \(k\).

• primitive_poly (galois.Poly, optional) – 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 (int, galois.Poly, optional) – 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, optional) – Optionally specify if the encoding should be systematic, meaning the codeword is the message with parity appended. The default is True.

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([1, 1, 1, 1, 0, 0, 0], order=2)

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

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

Methods

decode(codeword[, errors])	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 \(\mathrm{GF}(2^m)\) 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.

decode(codeword, errors=False)

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

Parameters

• codeword (numpy.ndarray, galois.FieldArray) – 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.FieldArray – The decoded message as either a \(k\)-length vector or \((N, k)\) matrix.

• int, np.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

Decode a single codeword.

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

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

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

# Corrupt the first bit in the codeword
In [4]: c[0] ^= 1

In [5]: dec_m = bch.decode(c); dec_m
Out[5]: GF([0, 0, 0, 1, 1, 1, 1], order=2)

In [6]: np.array_equal(dec_m, m)
Out[6]: True

# Instruct the decoder to return the number of corrected bit errors
In [7]: dec_m, N = bch.decode(c, errors=True); dec_m, N
Out[7]: (GF([0, 0, 0, 1, 1, 1, 1], order=2), 1)

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

Decode a single, shortened codeword.

In [9]: m = galois.GF2.Random(bch.k - 3); m
Out[9]: GF([0, 0, 1, 1], order=2)

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

# Corrupt the first bit in the codeword
In [11]: c[0] ^= 1

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

In [13]: np.array_equal(dec_m, m)
Out[13]: True

Decode a matrix of codewords.

In [14]: m = galois.GF2.Random((5, bch.k)); m
Out[14]: 
GF([[0, 0, 1, 0, 1, 1, 1],
    [1, 0, 0, 1, 1, 1, 1],
    [0, 0, 0, 1, 0, 0, 1],
    [0, 1, 1, 1, 0, 1, 0],
    [1, 0, 0, 1, 0, 1, 1]], order=2)

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

# Corrupt the first bit in each codeword
In [16]: c[:,0] ^= 1

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

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

# Instruct the decoder to return the number of corrected bit errors
In [19]: dec_m, N = bch.decode(c, errors=True); dec_m, N
Out[19]: 
(GF([[0, 0, 1, 0, 1, 1, 1],
     [1, 0, 0, 1, 1, 1, 1],
     [0, 0, 0, 1, 0, 0, 1],
     [0, 1, 1, 1, 0, 1, 0],
     [1, 0, 0, 1, 0, 1, 1]], order=2),
 array([1, 1, 1, 1, 1]))

In [20]: np.array_equal(dec_m, m)
Out[20]: 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 (numpy.ndarray, galois.FieldArray) – 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

bool, numpy.ndarray

Examples

Detect errors in a valid codeword.

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

# The minimum distance of the code
In [2]: bch.d
Out[2]: 5

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

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

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

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

# Corrupt the first `d - 1` bits in the codeword
In [6]: c[0:bch.d - 1] ^= 1

In [7]: bch.detect(c)
Out[7]: True

encode(message, parity_only=False)

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

Parameters

• message (numpy.ndarray, galois.FieldArray) – 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, optional) – 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

numpy.ndarray, galois.FieldArray

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

Encode a single codeword.

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

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

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

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

Encode a single, shortened codeword.

In [5]: m = galois.GF2.Random(bch.k - 3); m
Out[5]: GF([0, 0, 1, 0], order=2)

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

Encode a matrix of codewords.

In [7]: m = galois.GF2.Random((5, bch.k)); m
Out[7]: 
GF([[1, 0, 1, 1, 1, 1, 0],
    [1, 1, 1, 1, 1, 1, 1],
    [0, 0, 1, 0, 1, 1, 0],
    [0, 1, 1, 0, 0, 1, 1],
    [1, 0, 0, 0, 1, 0, 1]], order=2)

In [8]: c = bch.encode(m); c
Out[8]: 
GF([[1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0],
    [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],
    [0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1],
    [0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0],
    [1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1]], order=2)

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

property G

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)

Type

galois.GF2

property H

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)

Type

galois.FieldArray

property d

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

Type

int

property field

The Galois field \(\mathrm{GF}(2^m)\) 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.properties)
GF(2^4):
  characteristic: 2
  degree: 4
  order: 16
  irreducible_poly: x^4 + x + 1
  is_primitive_poly: True
  primitive_element: x

Type

galois.FieldClass

property generator_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)

Type

galois.Poly

property 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. \(\alpha, \alpha^2, \dots, \alpha^{2t}\).

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)

Type

bool

property is_primitive

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

Type

bool

property k

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

Type

int

property n

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

Type

int

property roots

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)

Type

galois.FieldArray

property systematic

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

Type

bool

property t

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

Type

int