# 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([15,  2,  5,  5, 13,  9,  6,  6,  4], order=2^4)

In [4]: c = rs.encode(m); c
Out[4]:
GF([15,  2,  5,  5, 13,  9,  6,  6,  4,  7,  2,  6, 10,  7,  7],
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([15,  2,  5,  5, 13,  9,  6,  6,  4], 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([15,  2,  5,  5, 13,  9,  6,  6,  4], 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.

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.
__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 (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.

Returns

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

Return type

galois.ReedSolomon

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([11, 14, 12,  1,  4,  8,  5,  9,  1], order=2^4)

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

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

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

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([15, 14, 13, 14, 14, 10, 15,  7,  0], order=2^4)

In [5]: c = rs.encode(m); c
Out[5]:
GF([15, 14, 13, 14, 14, 10, 15,  7,  0, 13,  8, 11,  6, 13,  8],
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

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,  5,  7,  7,  1, 11,  7, 12,  5], order=2^4)

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

In [5]: p = rs.encode(m, parity_only=True); p
Out[5]: GF([ 7,  5,  4, 14,  2,  3], order=2^4)


Encode a single, shortened codeword.

In [6]: m = GF.Random(rs.k - 4); m
Out[6]: GF([11,  3, 14,  0, 14], order=2^4)

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


Encode a matrix of codewords.

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

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

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