galois.ReedSolomon.detect(codeword: ArrayLike)

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

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 detect().

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([ 2,  1, 12, 13,  2,  8,  6,  8, 12], order=2^4)

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

In [8]: c[0:rs.d - 1] += e; c
Out[8]: GF([ 3, 11, 10,  7,  3,  1,  6,  8, 12,  6, 14, 15, 11,  9,  6], 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([ 7,  8,  7,  0, 11], order=2^4)

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

In [17]: c[0:rs.d - 1] += e; c
Out[17]: GF([ 9,  3, 10,  3,  7,  1, 12,  6,  4,  4, 13], 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([[ 4,  4,  3,  8,  9,  5, 11,  0, 13],
[12,  8,  5,  1,  8,  2,  2, 10, 14],
[ 1,  6,  9,  9,  6,  7,  8,  0,  4]], order=2^4)

In [22]: c = rs.encode(m); c
Out[22]:
GF([[ 4,  4,  3,  8,  9,  5, 11,  0, 13,  8,  7, 15,  1,  3, 11],
[12,  8,  5,  1,  8,  2,  2, 10, 14,  7,  1, 11,  3, 11,  6],
[ 1,  6,  9,  9,  6,  7,  8,  0,  4,  4,  0,  7,  7,  3, 14]],
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([[ 3,  4,  3,  8,  9,  5, 11,  0, 13,  8,  7, 15,  1,  3, 11],
[13,  2,  5,  1,  8,  2,  2, 10, 14,  7,  1, 11,  3, 11,  6],
[ 8, 12, 10,  2,  4,  1,  8,  0,  4,  4,  0,  7,  7,  3, 14]],
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([[ 0,  1,  8, 15,  9],
[ 2,  9, 10,  5, 11],
[ 1, 15,  4,  8, 14]], order=2^4)

In [33]: c = rs.encode(m); c
Out[33]:
GF([[ 0,  1,  8, 15,  9,  6, 15,  0, 15,  9, 12],
[ 2,  9, 10,  5, 11,  8, 14, 10,  7,  1,  5],
[ 1, 15,  4,  8, 14,  0,  0,  4,  7, 15, 11]], 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([[ 4,  1,  8, 15,  9,  6, 15,  0, 15,  9, 12],
[ 3,  0, 10,  5, 11,  8, 14, 10,  7,  1,  5],
[ 0, 10,  6,  2, 12, 15,  0,  4,  7, 15, 11]], order=2^4)

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