# galois.BCH¶

class galois.BCH(n: int, k: int, primitive_poly: = None, primitive_element: = None, systematic: bool = 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([1, 1, 1, 1, 1, 0, 1], order=2)

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

 A terse representation of the BCH code. A formatted string with relevant properties of the BCH code.

Methods

 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 FieldArray subclass 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: int, k: int, primitive_poly: = None, primitive_element: = None, systematic: bool = True)

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

Parameters
n

The codeword size $$n$$, must be $$n = 2^m - 1$$.

k

The message size $$k$$.

primitive_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 matlab_primitive_poly().

primitive_element

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 primitive_element().

systematic

Optionally specify if the encoding should be systematic, meaning the codeword is the message with parity appended. The default is True.

__repr__() str

A terse representation of the BCH code.

Examples

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

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

__str__() str

A formatted string with relevant properties of the BCH code.

Examples

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

In [2]: print(bch)
BCH Code:
[n, k, d]: [15, 7, 5]
field: GF(2)
generator_poly: x^8 + x^7 + x^6 + x^4 + 1
is_primitive: True
is_narrow_sense: True
systematic: True
t: 2

decode(codeword: , errors: Literal[False] = False)
decode(codeword: , errors: Literal[True])

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

Parameters
codeword

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

Optionally specify whether to return the number of corrected errors. The default is False.

Returns

• The decoded message as either a $$k$$-length vector or $$(N, k)$$ matrix.

• 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

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

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


Decode the codeword and recover the message.

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

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


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

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


Decode the codeword and recover the message.

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

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


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

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


Decode the codeword and recover the message.

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

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


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

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


Decode the codeword and recover the message.

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

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.

Examples

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

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

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


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

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

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


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

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

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


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

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

encode(message: , parity_only: bool = False)

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

Parameters
message

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

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.

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

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


Compute the parity bits only.

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

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

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


Compute the parity bits only.

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

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

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


Compute the parity bits only.

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

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

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


Compute the parity bits only.

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

property G : 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 : 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 : Type[FieldArray]

The FieldArray subclass 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]: galois.GF(2^4)

In [3]: print(bch.field)
<class 'galois.GF(2^4)'>

property generator_poly : 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 : 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: May 18, 2022