galois.BCH.detect(codeword: ArrayLike) → bool_ | ndarray

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

The \([n, k, d]_2\) BCH code has \(d_{min} \ge d\) minimum distance. It can detect up to \(d_{min}-1\) errors.

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 array indicating if errors were detected in the corresponding codeword True or not False.

Examples

Encode a single message using the \(\textrm{BCH}(15, 7)\) code.

In [1]: bch = galois.BCH(15, 7)

In [2]: GF = galois.GF(2)

In [3]: m = GF.Random(bch.k); m
Out[3]: GF([1, 0, 0, 0, 0, 0, 0], order=2)

In [4]: c = bch.encode(m); c
Out[4]: GF([1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0], order=2)

Detect no errors in the valid codeword.

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

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

In [6]: bch.d
Out[6]: 5

In [7]: c[0:bch.d - 1] ^= 1; c
Out[7]: GF([0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0], order=2)

In [8]: bch.detect(c)
Out[8]: True

Encode a single message using the shortened \(\textrm{BCH}(12, 4)\) code.

In [9]: bch = galois.BCH(15, 7)

In [10]: GF = galois.GF(2)

In [11]: m = GF.Random(bch.k - 3); m
Out[11]: GF([0, 0, 0, 1], order=2)

In [12]: c = bch.encode(m); c
Out[12]: GF([0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1], order=2)

Detect no errors in the valid codeword.

In [13]: bch.detect(c)
Out[13]: False

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

In [14]: bch.d
Out[14]: 5

In [15]: c[0:bch.d - 1] ^= 1; c
Out[15]: GF([1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1], order=2)

In [16]: bch.detect(c)
Out[16]: True

Encode a matrix of three messages using the \(\textrm{BCH}(15, 7)\) code.

In [17]: bch = galois.BCH(15, 7)

In [18]: GF = galois.GF(2)

In [19]: m = GF.Random((3, bch.k)); m
Out[19]: 
GF([[0, 1, 0, 1, 1, 1, 1],
    [0, 1, 1, 1, 0, 1, 1],
    [0, 0, 0, 1, 0, 1, 1]], order=2)

In [20]: c = bch.encode(m); c
Out[20]: 
GF([[0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1],
    [0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1],
    [0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1]], order=2)

Detect no errors in the valid codewords.

In [21]: bch.detect(c)
Out[21]: array([False, False, False])

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

In [22]: bch.d
Out[22]: 5

In [23]: c[0,0:1] ^= 1

In [24]: c[1,0:2] ^= 1

In [25]: c[2, 0:bch.d - 1] ^= 1

In [26]: c
Out[26]: 
GF([[1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1],
    [1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1],
    [1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1]], order=2)

In [27]: bch.detect(c)
Out[27]: array([ True,  True,  True])

Encode a matrix of three messages using the shortened \(\textrm{BCH}(12, 4)\) code.

In [28]: bch = galois.BCH(15, 7)

In [29]: GF = galois.GF(2)

In [30]: m = GF.Random((3, bch.k - 3)); m
Out[30]: 
GF([[0, 0, 1, 1],
    [0, 0, 1, 1],
    [1, 1, 0, 0]], order=2)

In [31]: c = bch.encode(m); c
Out[31]: 
GF([[0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0],
    [0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0],
    [1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1]], order=2)

Detect no errors in the valid codewords.

In [32]: bch.detect(c)
Out[32]: array([False, False, False])

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

In [33]: bch.d
Out[33]: 5

In [34]: c[0,0:1] ^= 1

In [35]: c[1,0:2] ^= 1

In [36]: c[2, 0:bch.d - 1] ^= 1

In [37]: c
Out[37]: 
GF([[1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0],
    [1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0],
    [0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1]], order=2)

In [38]: bch.detect(c)
Out[38]: array([ True,  True,  True])
Last update: Aug 27, 2022