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

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


Encode a message.

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

In : c = bch.encode(m); c
Out: 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 : c ^= 1

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

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

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

In : np.array_equal(dec_m, m)
Out: 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 : bch = galois.BCH(15, 7)

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

In : c = bch.encode(m); c
Out: 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 : c ^= 1

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

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

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

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


Decode a single, shortened codeword.

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

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

# Corrupt the first bit in the codeword
In : c ^= 1

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

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


Decode a matrix of codewords.

In : m = galois.GF2.Random((5, bch.k)); m
Out:
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 : c = bch.encode(m); c
Out:
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 : c[:,0] ^= 1

In : dec_m = bch.decode(c); dec_m
Out:
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 : np.array_equal(dec_m, m)
Out: True

# Instruct the decoder to return the number of corrected bit errors
In : dec_m, N = bch.decode(c, errors=True); dec_m, N
Out:
(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 : np.array_equal(dec_m, m)
Out: 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

Examples

Detect errors in a valid codeword.

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

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

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

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

In : bch.detect(c)
Out: False


Detect $$d_{min}-1$$ errors in a received codeword.

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

In : bch.detect(c)
Out: 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

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 : bch = galois.BCH(15, 7)

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

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

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


Encode a single, shortened codeword.

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

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


Encode a matrix of codewords.

In : m = galois.GF2.Random((5, bch.k)); m
Out:
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 : c = bch.encode(m); c
Out:
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 : p = bch.encode(m, parity_only=True); p
Out:
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 : bch = galois.BCH(15, 7); bch
Out: <BCH Code: [15, 7, 5] over GF(2)>

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

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

In : bch.d
Out: 5

Type

int

property field

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

Examples

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

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

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

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

# Evaluate the generator polynomial at its roots in GF(2^m)
In : bch.generator_poly(bch.roots, field=bch.field)
Out: 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 : bch = galois.BCH(15, 7); bch
Out: <BCH Code: [15, 7, 5] over GF(2)>

In : bch.is_narrow_sense
Out: True

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

In : bch.field.primitive_element**(np.arange(1, 2*bch.t + 1))
Out: 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 : bch = galois.BCH(15, 7); bch
Out: <BCH Code: [15, 7, 5] over GF(2)>

In : bch.is_primitive
Out: True

Type

bool

property k

The message size $$k$$ of the $$[n, k, d]_2$$ code

Examples

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

In : bch.k
Out: 7

Type

int

property n

The codeword size $$n$$ of the $$[n, k, d]_2$$ code

Examples

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

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

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

# Evaluate the generator polynomial at its roots in GF(2^m)
In : bch.generator_poly(bch.roots, field=bch.field)
Out: 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 : bch = galois.BCH(15, 7); bch
Out: <BCH Code: [15, 7, 5] over GF(2)>

In : bch.systematic
Out: True

Type

bool

property t

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

Examples

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

In : bch.t
Out: 2

Type

int