galois.ReedSolomon.detect(codeword: ArrayLike) → bool | ndarray

Detects if errors are present in the codeword \(\mathbf{c}\).

Shortened codes

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

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.

Returns:

A boolean scalar or \(N\)-length array indicating if errors were detected in the corresponding codeword.

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([15, 15,  9, 14, 14,  6, 13,  7, 14], order=2^4)

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

Detect no errors in the valid codeword.

In [5]: rs.detect(c)
Out[5]: False

Detect \(d_{min}-1\) errors in the codeword.

In [6]: rs.d
Out[6]: 7

In [7]: e = GF.Random(rs.d - 1, low=1); e
Out[7]: GF([ 3, 11,  9,  2,  9,  7], order=2^4)

In [8]: c[0:rs.d - 1] += e; c
Out[8]: GF([12,  4,  0, 12,  7,  1, 13,  7, 14,  9, 12, 13, 14,  7,  0], order=2^4)

In [9]: rs.detect(c)
Out[9]: True

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

In [10]: rs = galois.ReedSolomon(15, 9)

In [11]: GF = rs.field

In [12]: m = GF.Random(rs.k - 4); m
Out[12]: GF([11,  2,  5, 13, 15], order=2^4)

In [13]: c = rs.encode(m); c
Out[13]: GF([11,  2,  5, 13, 15,  2,  7, 10,  7,  6,  6], order=2^4)

Detect no errors in the valid codeword.

In [14]: rs.detect(c)
Out[14]: False

Detect \(d_{min}-1\) errors in the codeword.

In [15]: rs.d
Out[15]: 7

In [16]: e = GF.Random(rs.d - 1, low=1); e
Out[16]: GF([ 6,  3, 13, 13, 13,  3], order=2^4)

In [17]: c[0:rs.d - 1] += e; c
Out[17]: GF([13,  1,  8,  0,  2,  1,  7, 10,  7,  6,  6], order=2^4)

In [18]: rs.detect(c)
Out[18]: True

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

In [19]: rs = galois.ReedSolomon(15, 9)

In [20]: GF = rs.field

In [21]: m = GF.Random((3, rs.k)); m
Out[21]: 
GF([[ 2,  4,  4,  3,  4, 15, 12,  1,  2],
    [15,  0,  8, 12, 11, 14,  1,  1,  4],
    [10,  4,  6, 13,  5,  2,  3, 13,  6]], order=2^4)

In [22]: c = rs.encode(m); c
Out[22]: 
GF([[ 2,  4,  4,  3,  4, 15, 12,  1,  2,  1,  6, 12, 13,  8,  6],
    [15,  0,  8, 12, 11, 14,  1,  1,  4,  3,  4, 12,  0, 12,  5],
    [10,  4,  6, 13,  5,  2,  3, 13,  6, 15, 12,  4, 14,  1,  8]],
   order=2^4)

Detect no errors in the valid codewords.

In [23]: rs.detect(c)
Out[23]: array([False, False, False])

Detect one, two, and \(d_{min}-1\) errors in the codewords.

In [24]: rs.d
Out[24]: 7

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:rs.d - 1] += GF.Random(rs.d - 1, low=1)

In [28]: c
Out[28]: 
GF([[ 9,  4,  4,  3,  4, 15, 12,  1,  2,  1,  6, 12, 13,  8,  6],
    [ 4,  1,  8, 12, 11, 14,  1,  1,  4,  3,  4, 12,  0, 12,  5],
    [ 9, 11,  3,  2,  0, 10,  3, 13,  6, 15, 12,  4, 14,  1,  8]],
   order=2^4)

In [29]: rs.detect(c)
Out[29]: array([ True,  True,  True])

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

In [30]: rs = galois.ReedSolomon(15, 9)

In [31]: GF = rs.field

In [32]: m = GF.Random((3, rs.k - 4)); m
Out[32]: 
GF([[ 1,  8,  6, 14,  1],
    [11,  1,  4, 10, 15],
    [ 6,  5, 12,  7, 14]], order=2^4)

In [33]: c = rs.encode(m); c
Out[33]: 
GF([[ 1,  8,  6, 14,  1,  3, 12, 11, 12,  3,  8],
    [11,  1,  4, 10, 15,  4,  8, 11,  2,  1, 14],
    [ 6,  5, 12,  7, 14,  4,  8,  4, 14, 11,  2]], order=2^4)

Detect no errors in the valid codewords.

In [34]: rs.detect(c)
Out[34]: array([False, False, False])

Detect one, two, and \(d_{min}-1\) errors in the codewords.

In [35]: rs.d
Out[35]: 7

In [36]: c[0, 0:1] += GF.Random(1, low=1)

In [37]: c[1, 0:2] += GF.Random(2, low=1)

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

In [39]: c
Out[39]: 
GF([[ 5,  8,  6, 14,  1,  3, 12, 11, 12,  3,  8],
    [ 7, 12,  4, 10, 15,  4,  8, 11,  2,  1, 14],
    [11, 10, 11,  6,  4, 14,  8,  4, 14, 11,  2]], order=2^4)

In [40]: rs.detect(c)
Out[40]: array([ True,  True,  True])