galois.ReedSolomon.decode

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 $c$ into the message $m$ .

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 $m$ is a member of $GF (q)^{k}$ . The corresponding message polynomial $m (x)$ is a degree- $k$ polynomial over $GF (q)$ .

m = [m_{k - 1}, \dots, m_{1}, m_{0}] \in GF (q)^{k}

m (x) = m_{k - 1} x^{k - 1} + \dots + m_{1} x + m_{0} \in GF (q) [x]

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

c = [c_{n - 1}, \dots, c_{1}, c_{0}] \in GF (q)^{n}

c (x) = c_{n - 1} x^{n - 1} + \dots + c_{1} x + c_{0} \in GF (q) [x]

In decoding, the syndrome vector $s$ is computed by evaluating the received codeword $r$ at the roots $α^{c}, \dots, α^{c + d - 2}$ of the generator polynomial $g (x)$ . The equivalent polynomial operation computes the remainder of $r (x)$ by $g (x)$ .

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

s (x) = r (x) mod g (x) \in 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 $σ (x)$ and the corresponding error locations and values.

Examples¶

Vector Vector (shortened) Matrix Matrix (shortened)

Encode a single message using the $RS (15, 9)$ code.

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

In [2]: GF = rs.field

In [3]: m = GF.Random(rs.k); m
Out[3]: GF([ 7,  7,  5, 10, 10,  5, 11,  5, 12], order=2^4)

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

Corrupt $t$ symbols of the codeword.

In [5]: e = GF.Random(rs.t, low=1); e
Out[5]: GF([ 8, 13, 14], order=2^4)

In [6]: c[0:rs.t] += e; c
Out[6]: GF([15, 10, 11, 10, 10,  5, 11,  5, 12, 10,  5,  6, 13,  0,  5], order=2^4)

Decode the codeword and recover the message.

In [7]: d = rs.decode(c); d
Out[7]: GF([ 7,  7,  5, 10, 10,  5, 11,  5, 12], 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([ 7,  7,  5, 10, 10,  5, 11,  5, 12], order=2^4), 3)

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

Encode a single message using the shortened $RS (11, 5)$ 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([13,  9, 12, 15, 11], order=2^4)

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

Corrupt $t$ symbols of the codeword.

In [15]: e = GF.Random(rs.t, low=1); e
Out[15]: GF([4, 1, 3], order=2^4)

In [16]: c[0:rs.t] += e; c
Out[16]: GF([ 9,  8, 15, 15, 11, 10,  8, 11, 13,  5,  9], order=2^4)

Decode the codeword and recover the message.

In [17]: d = rs.decode(c); d
Out[17]: GF([13,  9, 12, 15, 11], 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([13,  9, 12, 15, 11], order=2^4), 3)

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

Encode a matrix of three messages using the $RS (15, 9)$ 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([[ 7, 11,  4, 10, 10,  5, 10,  3, 13],
    [13, 15, 14,  9,  1, 10,  8,  0,  2],
    [ 5,  4, 11,  8,  0, 15,  8,  3,  5]], order=2^4)

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

Decode the codeword and recover the message.

In [29]: d = rs.decode(c); d
Out[29]: 
GF([[ 7, 11,  4, 10, 10,  5, 10,  3, 13],
    [13, 15, 14,  9,  1, 10,  8,  0,  2],
    [ 5,  4, 11,  8,  0, 15,  8,  3,  5]], 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([[ 7, 11,  4, 10, 10,  5, 10,  3, 13],
     [13, 15, 14,  9,  1, 10,  8,  0,  2],
     [ 5,  4, 11,  8,  0, 15,  8,  3,  5]], 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 $RS (11, 5)$ 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([[ 5, 12,  7,  9, 15],
    [14,  5,  0, 14,  7],
    [10,  8,  9,  7, 11]], order=2^4)

In [36]: c = rs.encode(m); c
Out[36]: 
GF([[ 5, 12,  7,  9, 15, 13, 14,  2, 14,  0,  2],
    [14,  5,  0, 14,  7,  0, 10,  9, 11,  5, 11],
    [10,  8,  9,  7, 11, 13,  5,  3, 13,  3,  5]], 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, 12,  7,  9, 15, 13, 14,  2, 14,  0,  2],
    [ 1,  2,  0, 14,  7,  0, 10,  9, 11,  5, 11],
    [ 4,  0, 14,  7, 11, 13,  5,  3, 13,  3,  5]], order=2^4)

Decode the codeword and recover the message.

In [41]: d = rs.decode(c); d
Out[41]: 
GF([[ 5, 12,  7,  9, 15],
    [14,  5,  0, 14,  7],
    [10,  8,  9,  7, 11]], 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([[ 5, 12,  7,  9, 15],
     [14,  5,  0, 14,  7],
     [10,  8,  9,  7, 11]], order=2^4),
 array([1, 2, 3]))

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