galois.ReedSolomon¶

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

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

A \(\textrm{RS}(n, k)\) code is a \([n, k, d]_q\) linear block code.

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.

Parameters

n (int) – The codeword size \(n\), must be \(n = q - 1\).
k (int) – The message size \(k\). The error-correcting capability \(t\) is defined by \(n - k = 2t\).
c (int, optional) – The first consecutive power of \(\alpha\). The default is 1.
primitive_poly (galois.Poly, optional) – 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 (int, galois.Poly, optional) – 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 \(\mathrm{GF}(q)\), i.e. galois.primitive_element(p, m).
systematic (bool, optional) – Optionally specify if the encoding should be systematic, meaning the codeword is the message with parity appended. The default is True.

Examples

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

In [2]: GF = rs.field

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

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

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

In [6]: dec_m = rs.decode(c); dec_m
Out[6]: GF([ 7,  6, 15,  1,  6, 10, 14,  4,  1], 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([ 7,  6, 15,  1,  6, 10, 14,  4,  1], order=2^4), 1)

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

Constructors

Methods

`decode`(codeword[, errors])	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 \(\mathrm{GF}(q)\) that defines the Reed-Solomon code.
`generator_poly`	The generator polynomial \(g(x)\) whose roots are `ReedSolomon.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 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.

decode(codeword, errors=False)[source]¶

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

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 \(s\) is computed by \(\mathbf{s} = \mathbf{c}\mathbf{H}^T\), where \(\mathbf{H}\) is the parity-check matrix. The equivalent polynomial operation is \(s(x) = c(x)\ \textrm{mod}\ g(x)\). 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 \(\sigma(x)\) 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.

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.
int, np.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.

Examples

Decode a single codeword.

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

In [2]: GF = rs.field

In [3]: m = GF.Random(rs.k); m
Out[3]: GF([ 4, 12,  5,  5,  5,  2,  1,  4, 14], order=2^4)

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

# Corrupt the first symbol in the codeword
In [5]: c[0] += GF(13)

In [6]: dec_m = rs.decode(c); dec_m
Out[6]: GF([ 4, 12,  5,  5,  5,  2,  1,  4, 14], 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([ 4, 12,  5,  5,  5,  2,  1,  4, 14], order=2^4), 1)

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

Decode a single, shortened codeword.

In [10]: m = GF.Random(rs.k - 4); m
Out[10]: GF([ 0, 14,  0,  4,  8], order=2^4)

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

# Corrupt the first symbol in the codeword
In [12]: c[0] += GF(13)

In [13]: dec_m = rs.decode(c); dec_m
Out[13]: GF([ 0, 14,  0,  4,  8], order=2^4)

In [14]: np.array_equal(dec_m, m)
Out[14]: True

Decode a matrix of codewords.

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

In [16]: c = rs.encode(m); c
Out[16]: 
GF([[ 6,  3,  6,  8,  0,  3,  1, 12,  9,  0, 11,  2,  3,  7,  6],
    [14,  1,  1,  6,  8,  9,  9,  3,  6, 10,  3, 15, 10,  9,  6],
    [ 1, 15,  3,  7,  1,  6,  1,  2,  1,  1, 10,  0, 13,  7,  6],
    [15, 10, 15,  4,  4, 15,  7,  4, 14, 14, 15,  2,  9,  4, 14],
    [ 2,  6, 15, 11,  5, 12,  3,  2,  5, 11, 11, 11, 10, 15,  4]],
   order=2^4)

# Corrupt the first symbol in each codeword
In [17]: c[:,0] += GF(13)

In [18]: dec_m = rs.decode(c); dec_m
Out[18]: 
GF([[ 6,  3,  6,  8,  0,  3,  1, 12,  9],
    [14,  1,  1,  6,  8,  9,  9,  3,  6],
    [ 1, 15,  3,  7,  1,  6,  1,  2,  1],
    [15, 10, 15,  4,  4, 15,  7,  4, 14],
    [ 2,  6, 15, 11,  5, 12,  3,  2,  5]], order=2^4)

In [19]: np.array_equal(dec_m, m)
Out[19]: True

# Instruct the decoder to return the number of corrected symbol errors
In [20]: dec_m, N = rs.decode(c, errors=True); dec_m, N
Out[20]: 
(GF([[ 6,  3,  6,  8,  0,  3,  1, 12,  9],
     [14,  1,  1,  6,  8,  9,  9,  3,  6],
     [ 1, 15,  3,  7,  1,  6,  1,  2,  1],
     [15, 10, 15,  4,  4, 15,  7,  4, 14],
     [ 2,  6, 15, 11,  5, 12,  3,  2,  5]], order=2^4),
 array([1, 1, 1, 1, 1]))

In [21]: np.array_equal(dec_m, m)
Out[21]: True

detect(codeword)[source]¶

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 (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: bool, numpy.ndarray

Examples

Detect errors in a valid codeword.

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

In [2]: GF = rs.field

# The minimum distance of the code
In [3]: rs.d
Out[3]: 7

In [4]: m = GF.Random(rs.k); m
Out[4]: GF([13,  1,  5,  1, 10,  8, 13, 11,  8], order=2^4)

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

In [6]: rs.detect(c)
Out[6]: False

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

# Corrupt the first `d - 1` symbols in the codeword
In [7]: c[0:rs.d - 1] += GF(13)

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

encode(message, parity_only=False)[source]¶

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

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.

Parameters

message (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, optional) – 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

numpy.ndarray, galois.FieldArray

Examples

Encode a single codeword.

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

In [2]: GF = rs.field

In [3]: m = GF.Random(rs.k); m
Out[3]: GF([ 0, 10, 14, 10,  3,  4, 14,  3, 13], order=2^4)

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

In [5]: p = rs.encode(m, parity_only=True); p
Out[5]: GF([ 3, 13,  8,  4,  5, 10], order=2^4)

Encode a single, shortened codeword.

In [6]: m = GF.Random(rs.k - 4); m
Out[6]: GF([ 7,  1,  7,  7, 11], order=2^4)

In [7]: c = rs.encode(m); c
Out[7]: GF([ 7,  1,  7,  7, 11, 13, 10,  7,  2,  1, 15], order=2^4)

Encode a matrix of codewords.

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

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

In [10]: p = rs.encode(m, parity_only=True); p
Out[10]: 
GF([[ 6,  3,  6, 15,  4, 13],
    [12,  7,  1, 13, 13, 12],
    [15, 12, 12,  3,  3,  1],
    [ 2,  3, 10,  6, 15, 14],
    [11, 14, 14,  4,  2,  4]], order=2^4)

property G¶

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

Type: galois.FieldArray

property H¶

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

Type: galois.FieldArray

property c¶

The degree of the first consecutive root.

Type: int

property d¶

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, \(d_{min} = d\).

Type: int

property field¶

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

Type: galois.FieldClass

property generator_poly¶

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

Type: galois.Poly

property 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. \(\alpha, \alpha^2, \dots, \alpha^{2t - 1}\).

Type: bool

property k¶

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

Type: int

property n¶

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

Type: int

property roots¶

The roots of the generator polynomial. These are consecutive powers of \(\alpha\).

Type: galois.FieldArray

property systematic¶

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

Type: bool

property t¶

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

Type: int