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 \(\mathrm{GF}(q)\), see galois.primitive_element().

  • 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

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([10, 14, 14,  5, 10,  3, 15,  2,  3], order=2^4)

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

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

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 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.

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.

  • 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.

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 \(s\) 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.

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([12, 10,  4, 12,  7, 13,  0, 12, 11], order=2^4)

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

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

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

In [21]: np.array_equal(dec_m, m)
Out[21]: 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 (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,  4, 15,  6,  2,  5,  0,  3,  9], order=2^4)

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

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

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

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

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, 13,  4,  5, 14, 13,  3,  5,  2], order=2^4)

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

In [5]: p = rs.encode(m, parity_only=True); p
Out[5]: GF([1, 9, 2, 5, 7, 2], order=2^4)

Encode a single, shortened codeword.

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

In [7]: c = rs.encode(m); c
Out[7]: GF([ 6,  2, 13,  3, 10,  8,  7,  2,  2,  9,  3], order=2^4)

Encode a matrix of codewords.

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

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

In [10]: p = rs.encode(m, parity_only=True); p
Out[10]: 
GF([[10,  4,  9, 15,  4, 10],
    [15, 15,  6, 12,  5, 10],
    [14, 11,  5,  3,  3, 11],
    [14,  9, 10,  2, 15, 14],
    [ 2, 11, 10,  7,  1,  0]], order=2^4)
property G

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)
Type

galois.FieldArray

property H

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)
Type

galois.FieldArray

property c

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

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
Type

int

property field

The Galois field \(\mathrm{GF}(q)\) 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.properties)
GF(2^4):
  characteristic: 2
  degree: 4
  order: 16
  irreducible_poly: x^4 + x + 1
  is_primitive_poly: True
  primitive_element: x
Type

galois.FieldClass

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

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)
Type

bool

property k

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
Type

int

property n

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
Type

int

property roots

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)
Type

galois.FieldArray

property systematic

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
Type

bool

property t

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
Type

int