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.

Parameters

n (int) – The codeword size \(n\), must be \(n = 2^m - 1\).
k (int) – The message size \(k\).
primitive_poly (int, galois.Poly, optional) – Optionally specify the primitive polynomial that defines the extension field \(\mathrm{GF}(2^m)\). The default is None which uses the lexicographically-smallest primitive polynomial, i.e. galois.primitive_poly(2, m, method="smallest"). The use of the lexicographically-smallest primitive polynomial, as opposed to a Conway polynomial, is the default in textbooks, Matlab, and Octave.
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-smallest primitive element in \(\mathrm{GF}(2^m)\), i.e. galois.primitive_element(2, m).
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

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

In [2]: bch = galois.BCH(15, 7)

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 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

Methods

`decode`(codeword[, errors])	Decodes the BCH codeword into its message.
`encode`(message[, parity_only])	Encodes the message into a BCH codeword.

Attributes

`G`	The generator matrix \(G\) with shape \((k, n)\).
`H`	The parity-check matrix \(H\) with shape \((n-k, n)\).
`field`	The Galois field \(\mathrm{GF}(2^m)\) that defines the BCH code.
`generator_poly`	The generator polynomial \(g(x)\) whose roots are `BCH.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. \(\alpha, \alpha^2, \dots, \alpha^{2*t - 1}\).
`is_primitive`	Indicates if the BCH code is primitive, meaning \(n = 2^m - 1\).
`k`	The message size \(k\) of the \(\textrm{BCH}(n, k)\) code.
`n`	The codeword size \(n\) of the \(\textrm{BCH}(n, k)\) code.
`roots`	The 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)[source]¶

Decodes the BCH codeword into its message.

Parameters

codeword (np.ndarray, galois.FieldArray) – The codeword as either a \(n\)-length vector or \((N, n)\) matrix, where \(N\) is the number of codewords.
errors (bool, optional) – Optionally specify whether to return the nubmer of corrected errors.

Returns

np.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.

Examples

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

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

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

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

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

In [10]: m = galois.GF2.Random((5, bch.k)); m
Out[10]: 
GF([[1, 1, 0, 0, 0, 0, 1],
    [0, 0, 1, 0, 1, 0, 1],
    [1, 1, 1, 0, 0, 0, 1],
    [1, 1, 1, 1, 0, 0, 1],
    [0, 0, 1, 0, 0, 0, 1]], order=2)

In [11]: c = bch.encode(m); c
Out[11]: 
GF([[1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1],
    [0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1],
    [1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1],
    [1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0],
    [0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1]], order=2)

# Corrupt the first bit in the codeword
In [12]: c[:,0] ^= 1

In [13]: dec_m = bch.decode(c); dec_m
Out[13]: 
GF([[1, 1, 0, 0, 0, 0, 1],
    [0, 0, 1, 0, 1, 0, 1],
    [1, 1, 1, 0, 0, 0, 1],
    [1, 1, 1, 1, 0, 0, 1],
    [0, 0, 1, 0, 0, 0, 1]], order=2)

In [14]: np.array_equal(dec_m, m)
Out[14]: True

# Instruct the decoder to return the number of corrected bit errors
In [15]: dec_m, N = bch.decode(c, errors=True); dec_m, N
Out[15]: 
(GF([[1, 1, 0, 0, 0, 0, 1],
     [0, 0, 1, 0, 1, 0, 1],
     [1, 1, 1, 0, 0, 0, 1],
     [1, 1, 1, 1, 0, 0, 1],
     [0, 0, 1, 0, 0, 0, 1]], order=2),
 array([1, 1, 1, 1, 1]))

In [16]: np.array_equal(dec_m, m)
Out[16]: True

encode(message, parity_only=False)[source]¶

Encodes the message into a BCH codeword.

Parameters

message (np.ndarray, galois.FieldArray) – The message as either a \(k\)-length vector or \((N, k)\) matrix, where \(N\) is the number of messages.
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 either a \(n - k\)-length vector or \((N, n-k)\) matrix.

Return type

np.ndarray, galois.FieldArray

Examples

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

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

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

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

In [5]: bch = galois.BCH(15, 7)

In [6]: m = galois.GF2.Random((5, bch.k)); m
Out[6]: 
GF([[0, 1, 1, 0, 1, 1, 0],
    [1, 1, 0, 1, 0, 1, 0],
    [1, 0, 1, 1, 1, 1, 0],
    [1, 0, 0, 1, 1, 0, 1],
    [1, 1, 0, 1, 0, 1, 1]], order=2)

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

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

property G¶

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

Type: galois.GF2

property H¶

The parity-check matrix \(H\) with shape \((n-k, n)\).

Type: galois.FieldArray

property field¶

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

Type: galois.FieldClass

property generator_poly¶

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

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. \(\alpha, \alpha^2, \dots, \alpha^{2*t - 1}\).

Type: bool

property is_primitive¶

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

Type: bool

property k¶

The message size \(k\) of the \(\textrm{BCH}(n, k)\) code.

Type: int

property n¶

The codeword size \(n\) of the \(\textrm{BCH}(n, k)\) code.

Type: int

property roots¶

The roots of the generator polynomial. These are consecutive powers of \(\alpha\).

Type: galois.FieldArray

property systematic¶

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

Type: bool

property t¶

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

Type: int