galois.BCH.decode(codeword: ArrayLike, output: 'message' | 'codeword' = 'message', errors: False = False) → FieldArray
galois.BCH.decode(codeword: ArrayLike, output: 'message' | 'codeword' = 'message', errors: True = True) → tuple[FieldArray, int | np.ndarray]

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

Shortened codes

For the shortened \([n-s,\ k-s,\ d]\) code (only applicable for systematic codes), pass \(n-s\) symbols into decode() to return the \(k-s\)-symbol message.

Parameters
codeword: ArrayLike

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.

output: 'message' | 'codeword' = 'message'

Specify whether to return the error-corrected message or entire codeword. The default is "message".

errors: False = False
errors: True = True

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

Returns

  • If output="message", the error-corrected message as either a \(k\)-length vector or \((N, k)\) matrix. If output="codeword", the error-corrected codeword as either a \(n\)-length vector or \((N, n)\) matrix.

  • If errors=True, returns the number of corrected symbol errors as either a scalar or \(N\)-length array. 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 message vector \(\mathbf{m}\) is a member of \(\mathrm{GF}(q)^k\). The corresponding message polynomial \(m(x)\) is a degree-\(k\) polynomial over \(\mathrm{GF}(q)\).

\[\mathbf{m} = [m_{k-1},\ \dots,\ m_1,\ m_0] \in \mathrm{GF}(q)^k\]
\[m(x) = m_{k-1} x^{k-1} + \dots + m_1 x + m_0 \in \mathrm{GF}(q)[x]\]

The codeword vector \(\mathbf{c}\) is a member of \(\mathrm{GF}(q)^n\). The corresponding codeword polynomial \(c(x)\) is a degree-\(n\) polynomial over \(\mathrm{GF}(q)\). Each codeword polynomial \(c(x)\) is divisible by the generator polynomial \(g(x)\).

\[\mathbf{c} = [c_{n-1},\ \dots,\ c_1,\ c_0] \in \mathrm{GF}(q)^n\]
\[c(x) = c_{n-1} x^{n-1} + \dots + c_1 x + c_0 \in \mathrm{GF}(q)[x]\]

In decoding, the syndrome vector \(\mathbf{s}\) is computed by evaluating the received codeword \(\mathbf{r}\) in the extension field \(\mathrm{GF}(q^m)\) at the roots \(\alpha^c, \dots, \alpha^{c+d-2}\) of the generator polynomial \(g(x)\). The equivalent polynomial operation computes the remainder of \(r(x)\) by \(g(x)\) in the extension field \(\mathrm{GF}(q^m)\).

\[\mathbf{s} = [r(\alpha^c),\ \dots,\ r(\alpha^{c+d-2})] \in \mathrm{GF}(q^m)^{d-1}\]
\[s(x) = r(x)\ \textrm{mod}\ g(x) \in \mathrm{GF}(q^m)[x]\]

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.

Note

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

Examples

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

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

In [2]: GF = bch.field

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

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

Corrupt \(t\) symbols of the codeword.

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

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

Decode the codeword and recover the message.

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

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

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

Corrupt \(t\) symbols of the codeword.

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

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

Decode the codeword and recover the message.

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

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

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

Decode the codeword and recover the message.

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

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

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

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