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[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 \(\mathbf{r}\) 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([10,  6,  4,  6,  8,  8,  7,  3, 14], order=2^4)

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

Corrupt \(t\) symbols of the codeword.

In [5]: e = GF.Random(rs.t, low=1); e
Out[5]: GF([10, 13,  9], order=2^4)

In [6]: c[0:rs.t] += e; c
Out[6]: GF([ 0, 11, 13,  6,  8,  8,  7,  3, 14,  9, 14, 13,  1, 10,  5], order=2^4)

Decode the codeword and recover the message.

In [7]: d = rs.decode(c); d
Out[7]: GF([10,  6,  4,  6,  8,  8,  7,  3, 14], 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([10,  6,  4,  6,  8,  8,  7,  3, 14], 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,  0, 13,  5,  3], order=2^4)

In [14]: c = rs.encode(m); c
Out[14]: GF([ 3,  0, 13,  5,  3,  3,  8,  1,  6,  3,  7], order=2^4)

Corrupt \(t\) symbols of the codeword.

In [15]: e = GF.Random(rs.t, low=1); e
Out[15]: GF([10,  5,  9], order=2^4)

In [16]: c[0:rs.t] += e; c
Out[16]: GF([9, 5, 4, 5, 3, 3, 8, 1, 6, 3, 7], order=2^4)

Decode the codeword and recover the message.

In [17]: d = rs.decode(c); d
Out[17]: GF([ 3,  0, 13,  5,  3], 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,  0, 13,  5,  3], 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, 11,  6,  3, 12, 15,  2,  3, 15],
    [ 2,  2,  7,  0,  3,  0, 14,  7, 13],
    [13,  5,  2,  0,  3,  3,  5,  2,  1]], order=2^4)

In [24]: c = rs.encode(m); c
Out[24]: 
GF([[14, 11,  6,  3, 12, 15,  2,  3, 15,  3,  2, 12, 10,  9, 15],
    [ 2,  2,  7,  0,  3,  0, 14,  7, 13, 10,  0,  1,  6,  2,  0],
    [13,  5,  2,  0,  3,  3,  5,  2,  1,  2, 14, 10,  2, 12,  0]],
   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([[ 8, 11,  6,  3, 12, 15,  2,  3, 15,  3,  2, 12, 10,  9, 15],
    [14, 10,  7,  0,  3,  0, 14,  7, 13, 10,  0,  1,  6,  2,  0],
    [ 5,  8, 10,  0,  3,  3,  5,  2,  1,  2, 14, 10,  2, 12,  0]],
   order=2^4)

Decode the codeword and recover the message.

In [29]: d = rs.decode(c); d
Out[29]: 
GF([[14, 11,  6,  3, 12, 15,  2,  3, 15],
    [ 2,  2,  7,  0,  3,  0, 14,  7, 13],
    [13,  5,  2,  0,  3,  3,  5,  2,  1]], 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, 11,  6,  3, 12, 15,  2,  3, 15],
     [ 2,  2,  7,  0,  3,  0, 14,  7, 13],
     [13,  5,  2,  0,  3,  3,  5,  2,  1]], 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([[11, 15, 15, 14, 14],
    [ 4,  6,  0,  3, 14],
    [ 3,  4,  1,  9,  7]], order=2^4)

In [36]: c = rs.encode(m); c
Out[36]: 
GF([[11, 15, 15, 14, 14, 12,  2,  3, 15, 15,  5],
    [ 4,  6,  0,  3, 14, 15, 14,  6,  9,  9, 15],
    [ 3,  4,  1,  9,  7,  8, 11,  1, 12,  8,  9]], 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([[ 4, 15, 15, 14, 14, 12,  2,  3, 15, 15,  5],
    [ 3,  9,  0,  3, 14, 15, 14,  6,  9,  9, 15],
    [ 2,  5,  3,  9,  7,  8, 11,  1, 12,  8,  9]], order=2^4)

Decode the codeword and recover the message.

In [41]: d = rs.decode(c); d
Out[41]: 
GF([[11, 15, 15, 14, 14],
    [ 4,  6,  0,  3, 14],
    [ 3,  4,  1,  9,  7]], 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([[11, 15, 15, 14, 14],
     [ 4,  6,  0,  3, 14],
     [ 3,  4,  1,  9,  7]], order=2^4),
 array([1, 2, 3]))

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