galois.ReedSolomon¶

class galois.ReedSolomon(n, k, c=1, primitive_poly=None, primitive_element=None, systematic=True)¶

A general \(\textrm{RS}(n, k)\) code.

A \(\textrm{RS}(n, k)\) code is a \([n, k, d]_q\) linear block code with codeword size \(n\), message size \(k\), minimum distance \(d\), and symbols taken from an alphabet of size \(q\) (a prime power).

To create the shortened \(\textrm{RS}(n-s, k-s)\) code, construct the full-sized \(\textrm{RS}(n, k)\) code and then pass \(k-s\) symbols into encode() and \(n-s\) symbols into decode(). Shortened codes are only applicable for systematic codes.

Examples

Construct the Reed-Solomon code.

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

In [2]: GF = rs.field

Encode a message.

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

In [4]: c = rs.encode(m); c
Out[4]: GF([11, 15,  5,  0,  6,  1,  4,  2,  5, 11,  2,  9,  5,  6, 11], order=2^4)

Corrupt the codeword and decode the message.

# Corrupt the first symbol in the codeword
In [5]: c[0] ^= 13

In [6]: dec_m = rs.decode(c); dec_m
Out[6]: GF([11, 15,  5,  0,  6,  1,  4,  2,  5], order=2^4)

In [7]: np.array_equal(dec_m, m)
Out[7]: True

# Instruct the decoder to return the number of corrected symbol errors
In [8]: dec_m, N = rs.decode(c, errors=True); dec_m, N
Out[8]: (GF([11, 15,  5,  0,  6,  1,  4,  2,  5], order=2^4), 1)

In [9]: np.array_equal(dec_m, m)
Out[9]: True

Constructors

`__init__`(n, k[, c, primitive_poly, ...])	Constructs a general \(\textrm{RS}(n, k)\) code.

Special Methods

`__repr__`()	Return repr(self).
`__str__`()	Return str(self).

Methods

`decode`()	Decodes the Reed-Solomon codeword \(\mathbf{c}\) into the message \(\mathbf{m}\).
`detect`(codeword)	Detects if errors are present in the Reed-Solomon codeword \(\mathbf{c}\).
`encode`(message[, parity_only])	Encodes the message \(\mathbf{m}\) into the Reed-Solomon codeword \(\mathbf{c}\).

Attributes

`G`	The generator matrix \(\mathbf{G}\) with shape \((k, n)\).
`H`	The parity-check matrix \(\mathbf{H}\) with shape \((2t, n)\).
`c`	The degree of the first consecutive root.
`d`	The design distance \(d\) of the \([n, k, d]_q\) code.
`field`	The Galois field array class for the \(\mathrm{GF}(q)\) field that defines the Reed-Solomon code.
`generator_poly`	The generator polynomial \(g(x)\) whose roots are `roots`.
`is_narrow_sense`	Indicates if the Reed-Solomon code is narrow sense, meaning the roots of the generator polynomial are consecutive powers of \(\alpha\) starting at 1, i.e. \(\alpha, \alpha^2, \dots, \alpha^{2t - 1}\).
`k`	The message size \(k\) of the \([n, k, d]_q\) code.
`n`	The codeword size \(n\) of the \([n, k, d]_q\) code.
`roots`	The \(2t\) roots of the generator polynomial.
`systematic`	Indicates if the code is configured to return codewords in systematic form.
`t`	The error-correcting capability of the code.

__init__(n, k, c=1, primitive_poly=None, primitive_element=None, systematic=True)¶

Constructs a general \(\textrm{RS}(n, k)\) code.

Parameters

n : int: The codeword size \(n\), must be \(n = q - 1\) where \(q\) is a prime power.
k : int: The message size \(k\). The error-correcting capability \(t\) is defined by \(n - k = 2t\).
c : Optional[int]: The first consecutive power of \(\alpha\). The default is 1.
primitive_poly : Optional[galois.Poly]: Optionally specify the primitive polynomial that defines the extension field \(\mathrm{GF}(q)\). The default is None which uses Matlab’s default, see galois.matlab_primitive_poly(). Matlab tends to use the lexicographically-minimal primitive polynomial as a default instead of the Conway polynomial.
primitive_element : Optional[Union[int, galois.Poly]]: Optionally specify the primitive element \(\alpha\) of \(\mathrm{GF}(q)\) whose powers are roots of the generator polynomial \(g(x)\). The default is None which uses the lexicographically-minimal primitive element in \(\mathrm{GF}(q)\), see galois.primitive_element().
systematic : bool: Optionally specify if the encoding should be systematic, meaning the codeword is the message with parity appended. The default is True.

__repr__()¶: Return repr(self).

__str__()¶: Return str(self).

decode(codeword, errors=False)¶

Decodes the Reed-Solomon codeword \(\mathbf{c}\) into the message \(\mathbf{m}\).

Parameters

codeword : numpy.ndarray, galois.FieldArray: 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 : bool, optional: Optionally specify whether to return the nubmer of corrected errors.

Returns

numpy.ndarray, galois.FieldArray – The decoded message as either a \(k\)-length vector or \((N, k)\) matrix.
numpy.integer, numpy.ndarray – Optional return argument of the number of corrected symbol 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 \(\mathbf{c}\) is defined as \(\mathbf{c} = [c_{n-1}, \dots, c_1, c_0] \in \mathrm{GF}(q)^n\), which corresponds to the codeword polynomial \(c(x) = c_{n-1} x^{n-1} + \dots + c_1 x + c_0\). The message vector \(\mathbf{m}\) is defined as \(\mathbf{m} = [m_{k-1}, \dots, m_1, m_0] \in \mathrm{GF}(q)^k\), 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 \(\mathbf{s}\) is computed by \(\mathbf{s} = \mathbf{c}\mathbf{H}^T\), where \(\mathbf{H}\) is the parity-check matrix. The equivalent polynomial operation is the codeword polynomial evaluated at each root of the generator polynomial, i.e. \(\mathbf{s} = [c(\alpha^{c}), c(\alpha^{c+1}), \dots, c(\alpha^{c+2t-1})]\). 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 \(\sigma(x)\) and the corresponding error locations and values.

For the shortened \(\textrm{RS}(n-s, k-s)\) code (only applicable for systematic codes), pass \(n-s\) symbols into decode() to return the \(k-s\)-symbol message.

Examples

Vector

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([ 0, 14, 15, 10,  7, 12, 10, 11,  4], order=2^4)

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

Corrupt \(t\) symbols of the codeword.

In [5]: e = GF.Random(rs.t, low=1); e
Out[5]: GF([15,  6,  8], order=2^4)

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

Decode the codeword and recover the message.

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

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

Vector (shortened)

Encode a single message using the shortened \(\textrm{RS}(11, 5)\) 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([ 9,  3,  8,  5, 12], order=2^4)

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

Corrupt \(t\) symbols of the codeword.

In [15]: e = GF.Random(rs.t, low=1); e
Out[15]: GF([1, 1, 9], order=2^4)

In [16]: c[0:rs.t] += e; c
Out[16]: GF([ 8,  2,  1,  5, 12,  0,  1, 12,  0,  2, 10], order=2^4)

Decode the codeword and recover the message.

In [17]: d = rs.decode(c); d
Out[17]: GF([ 9,  3,  8,  5, 12], 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([ 9,  3,  8,  5, 12], order=2^4), 3)

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

Matrix

Encode a matrix of three messages using the \(\textrm{RS}(15, 9)\) 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([[15,  6,  7,  8, 15, 10, 13,  2, 15],
    [ 5,  6, 13,  8,  0,  8, 14,  8,  7],
    [14, 14,  3, 12, 11,  6,  1, 14,  3]], order=2^4)

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

Decode the codeword and recover the message.

In [29]: d = rs.decode(c); d
Out[29]: 
GF([[15,  6,  7,  8, 15, 10, 13,  2, 15],
    [ 5,  6, 13,  8,  0,  8, 14,  8,  7],
    [14, 14,  3, 12, 11,  6,  1, 14,  3]], 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([[15,  6,  7,  8, 15, 10, 13,  2, 15],
     [ 5,  6, 13,  8,  0,  8, 14,  8,  7],
     [14, 14,  3, 12, 11,  6,  1, 14,  3]], order=2^4),
 array([1, 2, 3]))

In [32]: np.array_equal(d, m)
Out[32]: True

Matrix (shortened)

Encode a matrix of three messages using the shortened \(\textrm{RS}(11, 5)\) 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([[ 9, 12, 13, 12, 12],
    [10,  5, 12,  5, 13],
    [10, 11, 10,  9,  2]], order=2^4)

In [36]: c = rs.encode(m); c
Out[36]: 
GF([[ 9, 12, 13, 12, 12,  2, 14, 12, 10,  5,  6],
    [10,  5, 12,  5, 13,  7,  0, 10,  4,  3,  7],
    [10, 11, 10,  9,  2, 11, 15,  6,  4,  0,  9]], 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([[ 7, 12, 13, 12, 12,  2, 14, 12, 10,  5,  6],
    [ 9,  3, 12,  5, 13,  7,  0, 10,  4,  3,  7],
    [ 8,  6, 15,  9,  2, 11, 15,  6,  4,  0,  9]], order=2^4)

Decode the codeword and recover the message.

In [41]: d = rs.decode(c); d
Out[41]: 
GF([[ 9, 12, 13, 12, 12],
    [10,  5, 12,  5, 13],
    [10, 11, 10,  9,  2]], 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([[ 9, 12, 13, 12, 12],
     [10,  5, 12,  5, 13],
     [10, 11, 10,  9,  2]], order=2^4),
 array([1, 2, 3]))

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

detect(codeword)¶

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

The \([n, k, d]_q\) Reed-Solomon code has \(d_{min} = d\) minimum distance. It can detect up to \(d_{min}-1\) errors.

Parameters

codeword : Union[numpy.ndarray, galois.FieldArray]: 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.

Return type

Union[numpy.bool_, numpy.ndarray]

Examples

Vector

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([ 5,  9,  3, 15,  6,  0,  0,  1,  4], order=2^4)

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

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

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

Vector (shortened)

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

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

In [17]: c[0:rs.d - 1] += e; c
Out[17]: GF([ 5,  9, 14,  4,  1,  6,  7, 15, 15,  8,  5], order=2^4)

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

Matrix

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([[ 3,  2,  2,  7,  0,  7,  8, 15,  7],
    [ 8,  4,  3,  1,  1,  1,  9, 15,  2],
    [11,  8, 14,  8, 11,  8,  1,  5,  0]], order=2^4)

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

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

Matrix (shortened)

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([[ 6,  9,  3, 12,  1],
    [12,  6,  2, 11, 12],
    [ 6,  0, 14,  5, 15]], order=2^4)

In [33]: c = rs.encode(m); c
Out[33]: 
GF([[ 6,  9,  3, 12,  1,  8,  4,  7,  3,  9,  3],
    [12,  6,  2, 11, 12, 15,  1,  6,  5,  1,  0],
    [ 6,  0, 14,  5, 15,  8,  4, 15, 13,  5,  8]], 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([[ 2,  9,  3, 12,  1,  8,  4,  7,  3,  9,  3],
    [ 4,  3,  2, 11, 12, 15,  1,  6,  5,  1,  0],
    [14, 11, 13,  8,  0,  0,  4, 15, 13,  5,  8]], order=2^4)

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

encode(message, parity_only=False)¶

Encodes the message \(\mathbf{m}\) into the Reed-Solomon codeword \(\mathbf{c}\).

Parameters

message : Union[numpy.ndarray, galois.FieldArray]: The message as either a \(k\)-length vector or \((N, k)\) matrix, where \(N\) is the number of messages. For systematic codes, message lengths less than \(k\) may be provided to produce shortened codewords.
parity_only : bool: Optionally specify whether to return only the parity symbols. This only applies to systematic codes. The default is False.

Returns

The codeword as either a \(n\)-length vector or \((N, n)\) matrix. The return type matches the message type. If parity_only=True, the parity symbols are returned as either a \(n - k\)-length vector or \((N, n-k)\) matrix.

Return type

Union[numpy.ndarray, galois.FieldArray]

Notes

The message vector \(\mathbf{m}\) is defined as \(\mathbf{m} = [m_{k-1}, \dots, m_1, m_0] \in \mathrm{GF}(q)^k\), which corresponds to the message polynomial \(m(x) = m_{k-1} x^{k-1} + \dots + m_1 x + m_0\). The codeword vector \(\mathbf{c}\) is defined as \(\mathbf{c} = [c_{n-1}, \dots, c_1, c_0] \in \mathrm{GF}(q)^n\), which corresponds to the codeword polynomial \(c(x) = c_{n-1} x^{n-1} + \dots + c_1 x + c_0\).

The codeword vector is computed from the message vector by \(\mathbf{c} = \mathbf{m}\mathbf{G}\), where \(\mathbf{G}\) is the generator matrix. The equivalent polynomial operation is \(c(x) = m(x)g(x)\). For systematic codes, \(\mathbf{G} = [\mathbf{I}\ |\ \mathbf{P}]\) such that \(\mathbf{c} = [\mathbf{m}\ |\ \mathbf{p}]\). And in polynomial form, \(p(x) = -(m(x) x^{n-k}\ \textrm{mod}\ g(x))\) with \(c(x) = m(x)x^{n-k} + p(x)\). For systematic and non-systematic codes, each codeword is a multiple of the generator polynomial, i.e. \(g(x)\ |\ c(x)\).

For the shortened \(\textrm{RS}(n-s, k-s)\) code (only applicable for systematic codes), pass \(k-s\) symbols into encode() to return the \(n-s\)-symbol codeword.

Examples

Vector

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([ 9,  7,  8,  6, 13, 10,  7,  9, 11], order=2^4)

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

Compute the parity symbols only.

In [5]: p = rs.encode(m, parity_only=True); p
Out[5]: GF([ 7,  8,  1,  4, 14,  1], order=2^4)

Vector (shortened)

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

In [6]: rs = galois.ReedSolomon(15, 9)

In [7]: GF = rs.field

In [8]: m = GF.Random(rs.k - 4); m
Out[8]: GF([11,  6,  9, 11, 15], order=2^4)

In [9]: c = rs.encode(m); c
Out[9]: GF([11,  6,  9, 11, 15, 10,  3,  2,  7, 15, 10], order=2^4)

Compute the parity symbols only.

In [10]: p = rs.encode(m, parity_only=True); p
Out[10]: GF([10,  3,  2,  7, 15, 10], order=2^4)

Matrix

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

In [11]: rs = galois.ReedSolomon(15, 9)

In [12]: GF = rs.field

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

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

Compute the parity symbols only.

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

Matrix (shortened)

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

In [16]: rs = galois.ReedSolomon(15, 9)

In [17]: GF = rs.field

In [18]: m = GF.Random((3, rs.k - 4)); m
Out[18]: 
GF([[ 3,  7,  8,  7, 13],
    [ 9,  9,  6,  7,  4],
    [ 0, 12, 10,  6,  3]], order=2^4)

In [19]: c = rs.encode(m); c
Out[19]: 
GF([[ 3,  7,  8,  7, 13,  0, 11,  9,  1,  0, 13],
    [ 9,  9,  6,  7,  4, 11, 14,  1,  1,  7, 15],
    [ 0, 12, 10,  6,  3,  3,  4, 12,  8,  7,  9]], order=2^4)

Compute the parity symbols only.

In [20]: p = rs.encode(m, parity_only=True); p
Out[20]: 
GF([[ 0, 11,  9,  1,  0, 13],
    [11, 14,  1,  1,  7, 15],
    [ 3,  4, 12,  8,  7,  9]], order=2^4)

property G : galois.FieldArray¶

The generator matrix \(\mathbf{G}\) with shape \((k, n)\).

Examples

In [1]: rs = galois.ReedSolomon(15, 9); rs
Out[1]: <Reed-Solomon Code: [15, 9, 7] over GF(2^4)>

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

property H : galois.FieldArray¶

The parity-check matrix \(\mathbf{H}\) with shape \((2t, n)\).

Examples

In [1]: rs = galois.ReedSolomon(15, 9); rs
Out[1]: <Reed-Solomon Code: [15, 9, 7] over GF(2^4)>

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

property c : int¶

The degree of the first consecutive root.

Examples

In [1]: rs = galois.ReedSolomon(15, 9); rs
Out[1]: <Reed-Solomon Code: [15, 9, 7] over GF(2^4)>

In [2]: rs.c
Out[2]: 1

property d : int¶

The design distance \(d\) of the \([n, k, d]_q\) code. The minimum distance of a Reed-Solomon code is exactly equal to the design distance, \(d_{min} = d\).

Examples

In [1]: rs = galois.ReedSolomon(15, 9); rs
Out[1]: <Reed-Solomon Code: [15, 9, 7] over GF(2^4)>

In [2]: rs.d
Out[2]: 7

property field : galois.FieldClass¶

The Galois field array class for the \(\mathrm{GF}(q)\) field that defines the Reed-Solomon code.

Examples

In [1]: rs = galois.ReedSolomon(15, 9); rs
Out[1]: <Reed-Solomon Code: [15, 9, 7] over GF(2^4)>

In [2]: rs.field
Out[2]: <class 'numpy.ndarray over GF(2^4)'>

In [3]: print(rs.field)
Galois Field:
  name: GF(2^4)
  characteristic: 2
  degree: 4
  order: 16
  irreducible_poly: x^4 + x + 1
  is_primitive_poly: True
  primitive_element: x

property generator_poly : galois.Poly¶

The generator polynomial \(g(x)\) whose roots are roots.

Examples

In [1]: rs = galois.ReedSolomon(15, 9); rs
Out[1]: <Reed-Solomon Code: [15, 9, 7] over GF(2^4)>

In [2]: rs.generator_poly
Out[2]: Poly(x^6 + 7x^5 + 9x^4 + 3x^3 + 12x^2 + 10x + 12, GF(2^4))

Evaluate the generator polynomial at its roots.

In [3]: rs.generator_poly(rs.roots)
Out[3]: GF([0, 0, 0, 0, 0, 0], order=2^4)

property is_narrow_sense : bool¶

Indicates if the Reed-Solomon code is narrow sense, meaning the roots of the generator polynomial are consecutive powers of \(\alpha\) starting at 1, i.e. \(\alpha, \alpha^2, \dots, \alpha^{2t - 1}\).

Examples

In [1]: rs = galois.ReedSolomon(15, 9); rs
Out[1]: <Reed-Solomon Code: [15, 9, 7] over GF(2^4)>

In [2]: rs.is_narrow_sense
Out[2]: True

In [3]: rs.roots
Out[3]: GF([ 2,  4,  8,  3,  6, 12], order=2^4)

In [4]: rs.field.primitive_element**(np.arange(1, 2*rs.t + 1))
Out[4]: GF([ 2,  4,  8,  3,  6, 12], order=2^4)

property k : int¶

The message size \(k\) of the \([n, k, d]_q\) code.

Examples

In [1]: rs = galois.ReedSolomon(15, 9); rs
Out[1]: <Reed-Solomon Code: [15, 9, 7] over GF(2^4)>

In [2]: rs.k
Out[2]: 9

property n : int¶

The codeword size \(n\) of the \([n, k, d]_q\) code.

Examples

In [1]: rs = galois.ReedSolomon(15, 9); rs
Out[1]: <Reed-Solomon Code: [15, 9, 7] over GF(2^4)>

In [2]: rs.n
Out[2]: 15

property roots : galois.FieldArray¶

The \(2t\) roots of the generator polynomial. These are consecutive powers of \(\alpha\), specifically \(\alpha^c, \alpha^{c+1}, \dots, \alpha^{c+2t-1}\).

Examples

In [1]: rs = galois.ReedSolomon(15, 9); rs
Out[1]: <Reed-Solomon Code: [15, 9, 7] over GF(2^4)>

In [2]: rs.roots
Out[2]: GF([ 2,  4,  8,  3,  6, 12], order=2^4)

Evaluate the generator polynomial at its roots.

In [3]: rs.generator_poly(rs.roots)
Out[3]: GF([0, 0, 0, 0, 0, 0], order=2^4)

property systematic : bool¶

Indicates if the code is configured to return codewords in systematic form.

Examples

In [1]: rs = galois.ReedSolomon(15, 9); rs
Out[1]: <Reed-Solomon Code: [15, 9, 7] over GF(2^4)>

In [2]: rs.systematic
Out[2]: True

property t : int¶

The error-correcting capability of the code. The code can correct \(t\) symbol errors in a codeword.

Examples

In [1]: rs = galois.ReedSolomon(15, 9); rs
Out[1]: <Reed-Solomon Code: [15, 9, 7] over GF(2^4)>

In [2]: rs.t
Out[2]: 3

Last update: Apr 03, 2022