-
galois.BCH.decode(codeword: ArrayLike, output: Literal[message] | Literal[codeword] =
'message'
, errors: False =False
) FieldArray -
galois.BCH.decode(codeword: ArrayLike, output: Literal[message] | Literal[codeword] =
'message'
, errors: True =True
) tuple[FieldArray, int | np.ndarray] Decodes the codeword \(\mathbf{c}\) into the message \(\mathbf{m}\).
- Parameters:¶
- codeword: ArrayLike¶
The codeword as either a \(n\)-length vector or \((N, n)\) matrix, where \(N\) is the number of codewords.
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.- output: Literal[message] | Literal[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. Ifoutput="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([0, 0, 0, 0, 1, 0, 1], order=2) In [4]: c = bch.encode(m); c Out[4]: GF([0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1], 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([1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1], order=2)
Decode the codeword and recover the message.
In [7]: d = bch.decode(c); d Out[7]: GF([0, 0, 0, 0, 1, 0, 1], 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, 0, 0, 0, 1, 0, 1], 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, 0, 0], order=2) In [14]: c = bch.encode(m); c Out[14]: GF([1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1], 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, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1], order=2)
Decode the codeword and recover the message.
In [17]: d = bch.decode(c); d Out[17]: GF([1, 1, 0, 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, 0, 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, 0, 0, 0], [0, 1, 0, 1, 0, 1, 1], [1, 1, 1, 1, 1, 0, 0]], order=2) In [24]: c = bch.encode(m); c Out[24]: GF([[1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1], [0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1], [1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1]], 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, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1], [1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1], [0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1]], order=2)
Decode the codeword and recover the message.
In [28]: d = bch.decode(c); d Out[28]: GF([[1, 0, 0, 1, 0, 0, 0], [0, 1, 0, 1, 0, 1, 1], [1, 1, 1, 1, 1, 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, 0, 0, 1, 0, 0, 0], [0, 1, 0, 1, 0, 1, 1], [1, 1, 1, 1, 1, 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 = bch.field In [34]: m = GF.Random((3, bch.k - 3)); m Out[34]: GF([[0, 0, 1, 1], [1, 0, 0, 0], [1, 1, 0, 1]], order=2) In [35]: c = bch.encode(m); c Out[35]: GF([[0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0], [1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1], [1, 1, 0, 1, 0, 0, 1, 0, 1, 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 [36]: c[1,0:1] ^= 1 In [37]: c[2,0:2] ^= 1 In [38]: c Out[38]: GF([[0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1], [0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0]], order=2)
Decode the codeword and recover the message.
In [39]: d = bch.decode(c); d Out[39]: GF([[0, 0, 1, 1], [1, 0, 0, 0], [1, 1, 0, 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([[0, 0, 1, 1], [1, 0, 0, 0], [1, 1, 0, 1]], order=2), array([0, 1, 2])) In [42]: np.array_equal(d, m) Out[42]: True