# galois.ReedSolomon¶

class galois.ReedSolomon(n: int, k: int, c: int = 1, primitive_poly: = None, primitive_element: = None, systematic: bool = 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([13, 11, 15,  4, 14,  3, 14,  1,  4], order=2^4)

In [4]: c = rs.encode(m); c
Out[4]: GF([13, 11, 15,  4, 14,  3, 14,  1,  4,  6, 15, 11,  2, 14,  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([13, 11, 15,  4, 14,  3, 14,  1,  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([13, 11, 15,  4, 14,  3, 14,  1,  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.

Special Methods

 A terse representation of the Reed-Solomon code. A formatted string with relevant properties of the Reed-Solomon code.

Methods

 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 FieldArray subclass 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: int, k: int, c: int = 1, primitive_poly: = None, primitive_element: = None, systematic: bool = True)

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

Parameters
n

The codeword size $$n$$, must be $$n = q - 1$$ where $$q$$ is a prime power.

k

The message size $$k$$. The error-correcting capability $$t$$ is defined by $$n - k = 2t$$.

c

The first consecutive power of $$\alpha$$. The default is 1.

primitive_poly

Optionally specify the primitive polynomial that defines the extension field $$\mathrm{GF}(q)$$. The default is None which uses Matlab’s default, see matlab_primitive_poly().

primitive_element

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 primitive_element().

systematic

Optionally specify if the encoding should be systematic, meaning the codeword is the message with parity appended. The default is True.

__repr__() str

A terse representation of the Reed-Solomon code.

Examples

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

In [2]: rs
Out[2]: <Reed-Solomon Code: [15, 9, 7] over GF(2^4)>

__str__() str

A formatted string with relevant properties of the Reed-Solomon code.

Examples

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

In [2]: print(rs)
Reed-Solomon Code:
[n, k, d]: [15, 9, 7]
field: GF(2^4)
generator_poly: x^6 + 7x^5 + 9x^4 + 3x^3 + 12x^2 + 10x + 12
is_narrow_sense: True
systematic: True
t: 3

decode(codeword: , errors: Literal[False] = False)
decode(codeword: , errors: Literal[True])

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

Parameters
codeword

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

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 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

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,  7,  8, 14,  5,  7,  6,  1,  6], order=2^4)

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


Corrupt $$t$$ symbols of the codeword.

In [5]: e = GF.Random(rs.t, low=1); e
Out[5]: GF([12, 15,  4], order=2^4)

In [6]: c[0:rs.t] += e; c
Out[6]: GF([ 9,  8, 12, 14,  5,  7,  6,  1,  6, 12,  5,  1, 10,  6, 12], order=2^4)


Decode the codeword and recover the message.

In [7]: d = rs.decode(c); d
Out[7]: GF([ 5,  7,  8, 14,  5,  7,  6,  1,  6], 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([ 5,  7,  8, 14,  5,  7,  6,  1,  6], order=2^4), 3)

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


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([ 0, 14, 15,  6, 10], order=2^4)

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


Corrupt $$t$$ symbols of the codeword.

In [15]: e = GF.Random(rs.t, low=1); e
Out[15]: GF([ 5, 10,  1], order=2^4)

In [16]: c[0:rs.t] += e; c
Out[16]: GF([ 5,  4, 14,  6, 10,  9, 13,  2, 12, 12,  4], order=2^4)


Decode the codeword and recover the message.

In [17]: d = rs.decode(c); d
Out[17]: GF([ 0, 14, 15,  6, 10], 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([ 0, 14, 15,  6, 10], order=2^4), 3)

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


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([[12,  7,  3,  2,  7,  8, 11, 10, 12],
[ 5,  4,  6, 14,  8,  4, 13,  4, 14],
[12,  4, 12, 14, 15, 14, 15, 14, 12]], order=2^4)

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


Decode the codeword and recover the message.

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

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


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([[ 7, 11,  4,  1, 14],
[ 0,  0,  2, 13,  2],
[ 5, 12,  5, 15,  8]], order=2^4)

In [36]: c = rs.encode(m); c
Out[36]:
GF([[ 7, 11,  4,  1, 14,  3, 14,  2,  3, 12,  3],
[ 0,  0,  2, 13,  2, 13, 11, 13,  7, 15,  1],
[ 5, 12,  5, 15,  8, 11,  2,  1,  7,  8, 15]], 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([[13, 11,  4,  1, 14,  3, 14,  2,  3, 12,  3],
[ 7, 11,  2, 13,  2, 13, 11, 13,  7, 15,  1],
[10,  1, 12, 15,  8, 11,  2,  1,  7,  8, 15]], order=2^4)


Decode the codeword and recover the message.

In [41]: d = rs.decode(c); d
Out[41]:
GF([[ 7, 11,  4,  1, 14],
[ 0,  0,  2, 13,  2],
[ 5, 12,  5, 15,  8]], 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([[ 7, 11,  4,  1, 14],
[ 0,  0,  2, 13,  2],
[ 5, 12,  5, 15,  8]], 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

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.

Examples

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,  1,  0, 10,  1,  7,  7, 10,  2], order=2^4)

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

In [8]: c[0:rs.d - 1] += e; c
Out[8]: GF([15, 13,  7, 12, 10, 11,  7, 10,  2,  0,  0,  7, 11,  1,  6], order=2^4)

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


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([ 8,  3,  1,  5, 10], order=2^4)

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

In [17]: c[0:rs.d - 1] += e; c
Out[17]: GF([ 4, 14,  6,  6,  2,  9,  4,  9,  2,  3,  9], order=2^4)

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


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([[12,  4,  5, 13, 11, 10,  4,  9,  1],
[15,  9, 13, 12,  3,  5,  4,  5, 14],
[ 3, 10,  3,  4, 12,  1, 13,  6, 13]], order=2^4)

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

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


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([[ 9,  5, 12, 15, 15],
[11, 11,  7,  7,  8],
[ 3,  1, 15,  5, 10]], order=2^4)

In [33]: c = rs.encode(m); c
Out[33]:
GF([[ 9,  5, 12, 15, 15,  0,  9,  5, 10, 10,  3],
[11, 11,  7,  7,  8,  7,  1,  9, 15,  8, 12],
[ 3,  1, 15,  5, 10,  8, 11, 15, 11,  6,  9]], 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([[ 1,  5, 12, 15, 15,  0,  9,  5, 10, 10,  3],
[ 4, 14,  7,  7,  8,  7,  1,  9, 15,  8, 12],
[11, 15, 11, 15, 12,  1, 11, 15, 11,  6,  9]], order=2^4)

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

encode(message: , parity_only: bool = False)

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

Parameters
message

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

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.

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

In [4]: c = rs.encode(m); c
Out[4]: GF([ 2,  0,  7,  2,  5,  3,  1,  4,  8,  3,  1,  8, 10,  5,  4], order=2^4)


Compute the parity symbols only.

In [5]: p = rs.encode(m, parity_only=True); p
Out[5]: GF([ 3,  1,  8, 10,  5,  4], order=2^4)


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([4, 6, 2, 7, 6], order=2^4)

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


Compute the parity symbols only.

In [10]: p = rs.encode(m, parity_only=True); p
Out[10]: GF([15, 13, 13, 11,  0, 14], order=2^4)


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,  4,  3, 13, 14,  9,  3],
[ 6, 14,  4,  2,  9,  8,  4,  3,  8],
[ 2,  2,  6, 10,  6, 15,  9,  7,  4]], order=2^4)

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


Compute the parity symbols only.

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


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([[ 8, 14,  7, 12,  6],
[ 7,  1,  3,  6,  9],
[ 4,  8,  4,  1,  6]], order=2^4)

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


Compute the parity symbols only.

In [20]: p = rs.encode(m, parity_only=True); p
Out[20]:
GF([[15,  6,  1,  4,  9,  4],
[ 4,  8, 10, 14,  4,  0],
[ 4,  9,  7, 12,  3, 15]], order=2^4)

property G : 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 : 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 : Type[FieldArray]

The FieldArray subclass 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]: galois.GF(2^4)

In [3]: print(rs.field)
<class 'galois.GF(2^4)'>

property generator_poly : 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 : 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: May 18, 2022