galois.ReedSolomon.decode(codeword: ArrayLike, output: 'message' | 'codeword' = 'message', errors: False = False) → FieldArray
galois.ReedSolomon.decode(codeword: ArrayLike, output: 'message' | 'codeword' = 'message', errors: True = True) → tuple[galois._fields._array.FieldArray, int | numpy.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: '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 \(\mathbf{r}\) 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 \(g(x)\).

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

\[s(x) = r(x)\ \textrm{mod}\ g(x) \in \mathrm{GF}(q)[x]\]

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.

Examples

Encode a single message using the \(\textrm{RS}(15, 9)\) code.

In [1]: rs = galois.ReedSolomon(15, 9)

In [2]: GF = rs.field

In [3]: m = GF.Random(rs.k); m
Out[3]: GF([ 1,  3,  7,  7, 11, 15,  5,  2,  3], order=2^4)

In [4]: c = rs.encode(m); c
Out[4]: GF([ 1,  3,  7,  7, 11, 15,  5,  2,  3,  3,  3, 11,  5,  8, 14], order=2^4)

Corrupt \(t\) symbols of the codeword.

In [5]: e = GF.Random(rs.t, low=1); e
Out[5]: GF([ 7, 13,  2], order=2^4)

In [6]: c[0:rs.t] += e; c
Out[6]: GF([ 6, 14,  5,  7, 11, 15,  5,  2,  3,  3,  3, 11,  5,  8, 14], order=2^4)

Decode the codeword and recover the message.

In [7]: d = rs.decode(c); d
Out[7]: GF([ 1,  3,  7,  7, 11, 15,  5,  2,  3], order=2^4)

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 = rs.decode(c, errors=True); d, e
Out[9]: (GF([ 1,  3,  7,  7, 11, 15,  5,  2,  3], order=2^4), 3)

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

Encode a single message using the shortened \(\textrm{RS}(11, 5)\) code.

In [11]: rs = galois.ReedSolomon(15, 9)

In [12]: GF = rs.field

In [13]: m = GF.Random(rs.k - 4); m
Out[13]: GF([ 3, 11,  1,  5,  1], order=2^4)

In [14]: c = rs.encode(m); c
Out[14]: GF([ 3, 11,  1,  5,  1,  7, 13, 13, 14,  3, 15], order=2^4)

Corrupt \(t\) symbols of the codeword.

In [15]: e = GF.Random(rs.t, low=1); e
Out[15]: GF([13,  9, 11], order=2^4)

In [16]: c[0:rs.t] += e; c
Out[16]: GF([14,  2, 10,  5,  1,  7, 13, 13, 14,  3, 15], order=2^4)

Decode the codeword and recover the message.

In [17]: d = rs.decode(c); d
Out[17]: GF([ 3, 11,  1,  5,  1], order=2^4)

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 = rs.decode(c, errors=True); d, e
Out[19]: (GF([ 3, 11,  1,  5,  1], order=2^4), 3)

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

Encode a matrix of three messages using the \(\textrm{RS}(15, 9)\) code.

In [21]: rs = galois.ReedSolomon(15, 9)

In [22]: GF = rs.field

In [23]: m = GF.Random((3, rs.k)); m
Out[23]: 
GF([[14,  8,  1, 10,  2, 15,  2, 11,  4],
    [12, 13,  7,  3,  3, 15,  2,  7,  5],
    [14,  9, 12,  7,  2, 10, 14,  8, 14]], order=2^4)

In [24]: c = rs.encode(m); c
Out[24]: 
GF([[14,  8,  1, 10,  2, 15,  2, 11,  4,  9, 10,  3, 13,  1, 14],
    [12, 13,  7,  3,  3, 15,  2,  7,  5,  3,  2,  5,  2,  7,  4],
    [14,  9, 12,  7,  2, 10, 14,  8, 14, 12,  5,  2,  6,  6, 15]],
   order=2^4)

Corrupt the codeword. Add one error to the first codeword, two to the second, and three to the third.

In [25]: c[0,0:1] += GF.Random(1, low=1)

In [26]: c[1,0:2] += GF.Random(2, low=1)

In [27]: c[2,0:3] += GF.Random(3, low=1)

In [28]: c
Out[28]: 
GF([[ 4,  8,  1, 10,  2, 15,  2, 11,  4,  9, 10,  3, 13,  1, 14],
    [ 8,  7,  7,  3,  3, 15,  2,  7,  5,  3,  2,  5,  2,  7,  4],
    [ 1,  4,  4,  7,  2, 10, 14,  8, 14, 12,  5,  2,  6,  6, 15]],
   order=2^4)

Decode the codeword and recover the message.

In [29]: d = rs.decode(c); d
Out[29]: 
GF([[14,  8,  1, 10,  2, 15,  2, 11,  4],
    [12, 13,  7,  3,  3, 15,  2,  7,  5],
    [14,  9, 12,  7,  2, 10, 14,  8, 14]], order=2^4)

In [30]: np.array_equal(d, m)
Out[30]: True

Decode the codeword, specifying the number of corrected errors, and recover the message.

In [31]: d, e = rs.decode(c, errors=True); d, e
Out[31]: 
(GF([[14,  8,  1, 10,  2, 15,  2, 11,  4],
     [12, 13,  7,  3,  3, 15,  2,  7,  5],
     [14,  9, 12,  7,  2, 10, 14,  8, 14]], order=2^4),
 array([1, 2, 3]))

In [32]: np.array_equal(d, m)
Out[32]: True

Encode a matrix of three messages using the shortened \(\textrm{RS}(11, 5)\) code.

In [33]: rs = galois.ReedSolomon(15, 9)

In [34]: GF = rs.field

In [35]: m = GF.Random((3, rs.k - 4)); m
Out[35]: 
GF([[12,  7, 15,  2,  4],
    [ 4,  1,  4,  0, 14],
    [ 0,  7, 10,  4, 13]], order=2^4)

In [36]: c = rs.encode(m); c
Out[36]: 
GF([[12,  7, 15,  2,  4,  0,  5, 12, 11,  5,  2],
    [ 4,  1,  4,  0, 14, 13,  8,  0,  5, 13, 11],
    [ 0,  7, 10,  4, 13,  3, 11, 12, 14, 15,  0]], order=2^4)

Corrupt the codeword. Add one error to the first codeword, two to the second, and three to the third.

In [37]: c[0,0:1] += GF.Random(1, low=1)

In [38]: c[1,0:2] += GF.Random(2, low=1)

In [39]: c[2,0:3] += GF.Random(3, low=1)

In [40]: c
Out[40]: 
GF([[ 9,  7, 15,  2,  4,  0,  5, 12, 11,  5,  2],
    [ 7,  8,  4,  0, 14, 13,  8,  0,  5, 13, 11],
    [ 7,  0,  6,  4, 13,  3, 11, 12, 14, 15,  0]], order=2^4)

Decode the codeword and recover the message.

In [41]: d = rs.decode(c); d
Out[41]: 
GF([[12,  7, 15,  2,  4],
    [ 4,  1,  4,  0, 14],
    [ 0,  7, 10,  4, 13]], order=2^4)

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

Decode the codeword, specifying the number of corrected errors, and recover the message.

In [43]: d, e = rs.decode(c, errors=True); d, e
Out[43]: 
(GF([[12,  7, 15,  2,  4],
     [ 4,  1,  4,  0, 14],
     [ 0,  7, 10,  4, 13]], order=2^4),
 array([1, 2, 3]))

In [44]: np.array_equal(d, m)
Out[44]: True