galois.BCH.decode

galois.BCH.decode(codeword: ndarray | GF2, errors: False = False) → ndarray | GF2

galois.BCH.decode(codeword: ndarray | GF2, errors: True) → Tuple[ndarray | GF2, integer | ndarray]

Decodes the BCH codeword $c$ into the message $m$ .

Parameters¶

codeword: ndarray | GF2¶: 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.
errors: False = False¶
errors: True: Optionally specify whether to return the number of corrected errors. The default is False.

Returns¶

The decoded message as either a $k$ -length vector or $(N, k)$ matrix.
Optional return argument of the number of corrected bit errors as either a scalar or $n$ -length vector. 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 codeword vector $c$ is defined as $c = [c_{n - 1}, \dots, c_{1}, c_{0}] \in GF (2)^{n}$ , which corresponds to the codeword polynomial $c (x) = c_{n - 1} x^{n - 1} + \dots + c_{1} x + c_{0}$ . The message vector $m$ is defined as $m = [m_{k - 1}, \dots, m_{1}, m_{0}] \in GF (2)^{k}$ , which corresponds to the message polynomial $m (x) = m_{k - 1} x^{k - 1} + \dots + m_{1} x + m_{0}$ .

In decoding, the syndrome vector $s$ is computed by $s = c H^{T}$ , where $H$ is the parity-check matrix. The equivalent polynomial operation is $s (x) = c (x) mod g (x)$ . 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.

For the shortened $BCH (n - s, k - s)$ code (only applicable for systematic codes), pass $n - s$ bits into decode() to return the $k - s$ -bit message.

Examples¶

Vector

Encode a single message using the $BCH (15, 7)$ 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([0, 0, 0, 1, 1, 1, 0], order=2)

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

Corrupt $t$ bits of the codeword.

In [5]: bch.t
Out[5]: 2

In [6]: c[0:bch.t] ^= 1; c
Out[6]: GF([1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0], order=2)

Decode the codeword and recover the message.

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

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 = bch.decode(c, errors=True); d, e
Out[9]: (GF([0, 0, 0, 1, 1, 1, 0], order=2), 2)

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

Vector (shortened)

Encode a single message using the shortened $BCH (12, 4)$ code.

In [11]: bch = galois.BCH(15, 7)

In [12]: GF = galois.GF(2)

In [13]: m = GF.Random(bch.k - 3); m
Out[13]: GF([0, 0, 1, 0], order=2)

In [14]: c = bch.encode(m); c
Out[14]: GF([0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1], order=2)

Corrupt $t$ bits of the codeword.

In [15]: bch.t
Out[15]: 2

In [16]: c[0:bch.t] ^= 1; c
Out[16]: GF([1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1], order=2)

Decode the codeword and recover the message.

In [17]: d = bch.decode(c); d
Out[17]: GF([0, 0, 1, 0], order=2)

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 = bch.decode(c, errors=True); d, e
Out[19]: (GF([0, 0, 1, 0], order=2), 2)

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

Matrix

Encode a matrix of three messages using the $BCH (15, 7)$ code.

In [21]: bch = galois.BCH(15, 7)

In [22]: GF = galois.GF(2)

In [23]: m = GF.Random((3, bch.k)); m
Out[23]: 
GF([[1, 1, 0, 1, 0, 0, 1],
    [0, 0, 1, 1, 0, 0, 1],
    [1, 0, 0, 0, 0, 0, 1]], order=2)

In [24]: c = bch.encode(m); c
Out[24]: 
GF([[1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0],
    [0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0],
    [1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1]], order=2)

Corrupt the codeword. Add zero errors to the first codeword, one to the second, and two to the third.

In [25]: c[1,0:1] ^= 1

In [26]: c[2,0:2] ^= 1

In [27]: c
Out[27]: 
GF([[1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0],
    [1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0],
    [0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1]], order=2)

Decode the codeword and recover the message.

In [28]: d = bch.decode(c); d
Out[28]: 
GF([[1, 1, 0, 1, 0, 0, 1],
    [0, 0, 1, 1, 0, 0, 1],
    [1, 0, 0, 0, 0, 0, 1]], order=2)

In [29]: np.array_equal(d, m)
Out[29]: True

Decode the codeword, specifying the number of corrected errors, and recover the message.

In [30]: d, e = bch.decode(c, errors=True); d, e
Out[30]: 
(GF([[1, 1, 0, 1, 0, 0, 1],
     [0, 0, 1, 1, 0, 0, 1],
     [1, 0, 0, 0, 0, 0, 1]], order=2),
 array([0, 1, 2]))

In [31]: np.array_equal(d, m)
Out[31]: True

Matrix (shortened)

Encode a matrix of three messages using the shortened $BCH (12, 4)$ code.

In [32]: bch = galois.BCH(15, 7)

In [33]: GF = galois.GF(2)

In [34]: m = GF.Random((3, bch.k - 3)); m
Out[34]: 
GF([[1, 1, 1, 1],
    [0, 0, 0, 0],
    [1, 0, 1, 1]], order=2)

In [35]: c = bch.encode(m); c
Out[35]: 
GF([[1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1],
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
    [1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1]], order=2)

Corrupt the codeword. Add zero errors to the first codeword, one to the second, and two to the third.

In [36]: c[1,0:1] ^= 1

In [37]: c[2,0:2] ^= 1

In [38]: c
Out[38]: 
GF([[1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1],
    [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
    [0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1]], order=2)

Decode the codeword and recover the message.

In [39]: d = bch.decode(c); d
Out[39]: 
GF([[1, 1, 1, 1],
    [0, 0, 0, 0],
    [1, 0, 1, 1]], order=2)

In [40]: np.array_equal(d, m)
Out[40]: True

Decode the codeword, specifying the number of corrected errors, and recover the message.

In [41]: d, e = bch.decode(c, errors=True); d, e
Out[41]: 
(GF([[1, 1, 1, 1],
     [0, 0, 0, 0],
     [1, 0, 1, 1]], order=2),
 array([0, 1, 2]))

In [42]: np.array_equal(d, m)
Out[42]: True