ReedSolomon codes and Generalized ReedSolomon codes¶
Given \(n\) different evaluation points \(\alpha_1, \dots, \alpha_n\) from some finite field \(F\), the corresponding ReedSolomon code (RS code) of dimension \(k\) is the set:
An RS code is often called “classical” if \(alpha_i = \alpha^{i1}\) and \(\alpha\) is a primitive \(n\)‘th root of unity.
More generally, given also \(n\) “column multipliers” \(\beta_1, \dots, \beta_n\), the corresponding Generalized ReedSolomon code (GRS code) of dimension \(k\) is the set:
Here is a list of all content related to GRS codes:
GeneralizedReedSolomonCode
, the class for GRS codesReedSolomonCode()
, function for constructing classical ReedSolomon codes.GRSEvaluationVectorEncoder
, an encoder with a vectorial message spaceGRSEvaluationPolynomialEncoder
, an encoder with a polynomial message spaceGRSBerlekampWelchDecoder
, a decoder which corrects errors using BerlekampWelch algorithmGRSGaoDecoder
, a decoder which corrects errors using Gao algorithmGRSErrorErasureDecoder
, a decoder which corrects both errors and erasuresGRSKeyEquationSyndromeDecoder
, a decoder which corrects errors using the key equation on syndrome polynomials

class
sage.coding.grs.
GRSBerlekampWelchDecoder
(code)¶ Bases:
sage.coding.decoder.Decoder
Decoder for (Generalized) ReedSolomon codes which uses BerlekampWelch decoding algorithm to correct errors in codewords.
This algorithm recovers the error locator polynomial by solving a linear system. See [HJ2004] pp. 5152 for details.
INPUT:
code
– a code associated to this decoder
EXAMPLES:
sage: F = GF(59) sage: n, k = 40, 12 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: D = codes.decoders.GRSBerlekampWelchDecoder(C) sage: D BerlekampWelch decoder for [40, 12, 29] ReedSolomon Code over GF(59)
Actually, we can construct the decoder from
C
directly:sage: D = C.decoder("BerlekampWelch") sage: D BerlekampWelch decoder for [40, 12, 29] ReedSolomon Code over GF(59)

decode_to_code
(r)¶ Correct the errors in
r
and returns a codeword.Note
If the code associated to
self
has the same length as its dimension,r
will be returned as is.INPUT:
r
– a vector of the ambient space ofself.code()
OUTPUT:
 a vector of
self.code()
EXAMPLES:
sage: F = GF(59) sage: n, k = 40, 12 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: D = codes.decoders.GRSBerlekampWelchDecoder(C) sage: c = C.random_element() sage: Chan = channels.StaticErrorRateChannel(C.ambient_space(), D.decoding_radius()) sage: y = Chan(c) sage: c == D.decode_to_code(y) True

decode_to_message
(r)¶ Decode
r
to an element in message space ofself
.Note
If the code associated to
self
has the same length as its dimension,r
will be unencoded as is. In that case, ifr
is not a codeword, the output is unspecified.INPUT:
r
– a codeword ofself
OUTPUT:
 a vector of
self
message space
EXAMPLES:
sage: F = GF(59) sage: n, k = 40, 12 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: D = codes.decoders.GRSBerlekampWelchDecoder(C) sage: c = C.random_element() sage: Chan = channels.StaticErrorRateChannel(C.ambient_space(), D.decoding_radius()) sage: y = Chan(c) sage: D.connected_encoder().unencode(c) == D.decode_to_message(y) True

decoding_radius
()¶ Return maximal number of errors that
self
can decode.OUTPUT:
 the number of errors as an integer
EXAMPLES:
sage: F = GF(59) sage: n, k = 40, 12 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: D = codes.decoders.GRSBerlekampWelchDecoder(C) sage: D.decoding_radius() 14

class
sage.coding.grs.
GRSErrorErasureDecoder
(code)¶ Bases:
sage.coding.decoder.Decoder
Decoder for (Generalized) ReedSolomon codes which is able to correct both errors and erasures in codewords.
Let \(C\) be a GRS code of length \(n\) and dimension \(k\). Considering \(y\) a codeword with at most \(t\) errors (\(t\) being the \(\left\lfloor \frac{d1}{2} \right\rfloor\) decoding radius), and \(e\) the erasure vector, this decoder works as follows:
 Puncture the erased coordinates which are identified in \(e\).
 Create a new GRS code of length \(n  w(e)\), where \(w\) is the Hamming weight function, and dimension \(k\).
 Use Gao decoder over this new code one the punctured word built on the first step.
 Recover the original message from the decoded word computed on the previous step.
 Encode this message using an encoder over \(C\).
INPUT:
code
– the associated code of this decoder
EXAMPLES:
sage: F = GF(59) sage: n, k = 40, 12 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: D = codes.decoders.GRSErrorErasureDecoder(C) sage: D ErrorErasure decoder for [40, 12, 29] ReedSolomon Code over GF(59)
Actually, we can construct the decoder from
C
directly:sage: D = C.decoder("ErrorErasure") sage: D ErrorErasure decoder for [40, 12, 29] ReedSolomon Code over GF(59)

decode_to_message
(word_and_erasure_vector)¶ Decode
word_and_erasure_vector
to an element in message space ofself
INPUT:
 word_and_erasure_vector – a tuple whose:
 first element is an element of the ambient space of the code
 second element is a vector over \(\GF{2}\) whose length is the same as the code’s
Note
If the code associated to
self
has the same length as its dimension,r
will be unencoded as is. If the number of erasures is exactly \(n  k\), where \(n\) is the length of the code associated toself
and \(k\) its dimension,r
will be returned as is. In either case, ifr
is not a codeword, the output is unspecified.INPUT:
word_and_erasure_vector
– a pair of vectors, where first element is a codeword ofself
and second element is a vector of GF(2) containing erasure positions
OUTPUT:
 a vector of
self
message space
EXAMPLES:
sage: F = GF(59) sage: n, k = 40, 12 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: D = codes.decoders.GRSErrorErasureDecoder(C) sage: c = C.random_element() sage: n_era = randint(0, C.minimum_distance()  2) sage: Chan = channels.ErrorErasureChannel(C.ambient_space(), D.decoding_radius(n_era), n_era) sage: y = Chan(c) sage: D.connected_encoder().unencode(c) == D.decode_to_message(y) True
 word_and_erasure_vector – a tuple whose:

decoding_radius
(number_erasures)¶ Return maximal number of errors that
self
can decode according to how many erasures it receives.INPUT:
number_erasures
– the number of erasures when we try to decode
OUTPUT:
 the number of errors as an integer
EXAMPLES:
sage: F = GF(59) sage: n, k = 40, 12 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: D = codes.decoders.GRSErrorErasureDecoder(C) sage: D.decoding_radius(5) 11
If we receive too many erasures, it returns an exception as codeword will be impossible to decode:
sage: D.decoding_radius(30) Traceback (most recent call last): ... ValueError: The number of erasures exceed decoding capability

class
sage.coding.grs.
GRSEvaluationPolynomialEncoder
(code, polynomial_ring=None)¶ Bases:
sage.coding.encoder.Encoder
Encoder for (Generalized) ReedSolomon codes which uses evaluation of polynomials to obtain codewords.
Let \(C\) be a GRS code of length \(n\) and dimension \(k\) over some finite field \(F\). We denote by \(\alpha_i\) its evaluations points and by \(\beta_i\) its column multipliers, where \(1 \leq i \leq n\). Let \(p\) be a polynomial of degree at most \(k1\) in \(F[x]\) be the message.
The encoding of \(m\) will be the following codeword:
\[(\beta_1 \times p(\alpha_1), \dots, \beta_n \times p(\alpha_n)).\]INPUT:
code
– the associated code of this encoderpolynomial_ring
– (default:None
) a polynomial ring to specify the message space ofself
, if needed; it is set to \(F[x]\) (where \(F\) is the base field ofcode
) if default value is kept
EXAMPLES:
sage: F = GF(59) sage: n, k = 40, 12 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: E = codes.encoders.GRSEvaluationPolynomialEncoder(C) sage: E Evaluation polynomialstyle encoder for [40, 12, 29] ReedSolomon Code over GF(59) sage: E.message_space() Univariate Polynomial Ring in x over Finite Field of size 59
Actually, we can construct the encoder from
C
directly:sage: E = C.encoder("EvaluationPolynomial") sage: E Evaluation polynomialstyle encoder for [40, 12, 29] ReedSolomon Code over GF(59)
We can also specify another polynomial ring:
sage: R = PolynomialRing(F, 'y') sage: E = C.encoder("EvaluationPolynomial", polynomial_ring=R) sage: E.message_space() Univariate Polynomial Ring in y over Finite Field of size 59

encode
(p)¶ Transform the polynomial
p
into a codeword ofcode()
.One can use the following shortcut to encode a word with an encoder
E
:E(word)
INPUT:
p
– a polynomial from the message space ofself
of degree less thanself.code().dimension()
OUTPUT:
 a codeword in associated code of
self
EXAMPLES:
sage: F = GF(11) sage: Fx.<x> = F[] sage: n, k = 10 , 5 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: E = C.encoder("EvaluationPolynomial") sage: p = x^2 + 3*x + 10 sage: c = E.encode(p); c (10, 3, 9, 6, 5, 6, 9, 3, 10, 8) sage: c in C True
If a polynomial of too high degree is given, an error is raised:
sage: p = x^10 sage: E.encode(p) Traceback (most recent call last): ... ValueError: The polynomial to encode must have degree at most 4
If
p
is not an element of the proper polynomial ring, an error is raised:sage: Qy.<y> = QQ[] sage: p = y^2 + 1 sage: E.encode(p) Traceback (most recent call last): ... ValueError: The value to encode must be in Univariate Polynomial Ring in x over Finite Field of size 11

message_space
()¶ Return the message space of
self
EXAMPLES:
sage: F = GF(11) sage: n, k = 10 , 5 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: E = C.encoder("EvaluationPolynomial") sage: E.message_space() Univariate Polynomial Ring in x over Finite Field of size 11

polynomial_ring
()¶ Return the message space of
self
EXAMPLES:
sage: F = GF(11) sage: n, k = 10 , 5 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: E = C.encoder("EvaluationPolynomial") sage: E.message_space() Univariate Polynomial Ring in x over Finite Field of size 11

unencode_nocheck
(c)¶ Return the message corresponding to the codeword
c
.Use this method with caution: it does not check if
c
belongs to the code, and if this is not the case, the output is unspecified. Instead, useunencode()
.INPUT:
c
– a codeword ofcode()
OUTPUT:
 a polynomial of degree less than
self.code().dimension()
EXAMPLES:
sage: F = GF(11) sage: n, k = 10 , 5 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: E = C.encoder("EvaluationPolynomial") sage: c = vector(F, (10, 3, 9, 6, 5, 6, 9, 3, 10, 8)) sage: c in C True sage: p = E.unencode_nocheck(c); p x^2 + 3*x + 10 sage: E.encode(p) == c True
Note that no error is thrown if
c
is not a codeword, and that the result is undefined:sage: c = vector(F, (11, 3, 9, 6, 5, 6, 9, 3, 10, 8)) sage: c in C False sage: p = E.unencode_nocheck(c); p 6*x^4 + 6*x^3 + 2*x^2 sage: E.encode(p) == c False

class
sage.coding.grs.
GRSEvaluationVectorEncoder
(code)¶ Bases:
sage.coding.encoder.Encoder
Encoder for (Generalized) ReedSolomon codes that encodes vectors into codewords.
Let \(C\) be a GRS code of length \(n\) and dimension \(k\) over some finite field \(F\). We denote by \(\alpha_i\) its evaluations points and by \(\beta_i\) its column multipliers, where \(1 \leq i \leq n\). Let \(m = (m_1, \dots, m_k)\), a vector over \(F\), be the message. We build a polynomial using the coordinates of \(m\) as coefficients:
\[p = \Sigma_{i=1}^{m} m_i \times x^i.\]The encoding of \(m\) will be the following codeword:
\[(\beta_1 \times p(\alpha_1), \dots, \beta_n \times p(\alpha_n)).\]INPUT:
code
– the associated code of this encoder
EXAMPLES:
sage: F = GF(59) sage: n, k = 40, 12 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: E = codes.encoders.GRSEvaluationVectorEncoder(C) sage: E Evaluation vectorstyle encoder for [40, 12, 29] ReedSolomon Code over GF(59)
Actually, we can construct the encoder from
C
directly:sage: E = C.encoder("EvaluationVector") sage: E Evaluation vectorstyle encoder for [40, 12, 29] ReedSolomon Code over GF(59)

generator_matrix
()¶ Return a generator matrix of
self
Considering a GRS code of length \(n\), dimension \(k\), with evaluation points \((\alpha_1, \dots, \alpha_n)\) and column multipliers \((\beta_1, \dots, \beta_n)\), its generator matrix \(G\) is built using the following formula:
\[G = [g_{i,j}], g_{i,j} = \beta_j \times \alpha_{j}^{i}.\]This matrix is a Vandermonde matrix.
EXAMPLES:
sage: F = GF(11) sage: n, k = 10, 5 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: E = codes.encoders.GRSEvaluationVectorEncoder(C) sage: E.generator_matrix() [1 1 1 1 1 1 1 1 1 1] [0 1 2 3 4 5 6 7 8 9] [0 1 4 9 5 3 3 5 9 4] [0 1 8 5 9 4 7 2 6 3] [0 1 5 4 3 9 9 3 4 5]

class
sage.coding.grs.
GRSGaoDecoder
(code)¶ Bases:
sage.coding.decoder.Decoder
Decoder for (Generalized) ReedSolomon codes which uses Gao decoding algorithm to correct errors in codewords.
Gao decoding algorithm uses early terminated extended Euclidean algorithm to find the error locator polynomial. See [Ga02] for details.
INPUT:
code
– the associated code of this decoder
EXAMPLES:
sage: F = GF(59) sage: n, k = 40, 12 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: D = codes.decoders.GRSGaoDecoder(C) sage: D Gao decoder for [40, 12, 29] ReedSolomon Code over GF(59)
Actually, we can construct the decoder from
C
directly:sage: D = C.decoder("Gao") sage: D Gao decoder for [40, 12, 29] ReedSolomon Code over GF(59)

decode_to_code
(r)¶ Correct the errors in
r
and returns a codeword.Note
If the code associated to
self
has the same length as its dimension,r
will be returned as is.INPUT:
r
– a vector of the ambient space ofself.code()
OUTPUT:
 a vector of
self.code()
EXAMPLES:
sage: F = GF(59) sage: n, k = 40, 12 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: D = codes.decoders.GRSGaoDecoder(C) sage: c = C.random_element() sage: Chan = channels.StaticErrorRateChannel(C.ambient_space(), D.decoding_radius()) sage: y = Chan(c) sage: c == D.decode_to_code(y) True

decode_to_message
(r)¶ Decode
r
to an element in message space ofself
.Note
If the code associated to
self
has the same length as its dimension,r
will be unencoded as is. In that case, ifr
is not a codeword, the output is unspecified.INPUT:
r
– a codeword ofself
OUTPUT:
 a vector of
self
message space
EXAMPLES:
sage: F = GF(59) sage: n, k = 40, 12 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: D = codes.decoders.GRSGaoDecoder(C) sage: c = C.random_element() sage: Chan = channels.StaticErrorRateChannel(C.ambient_space(), D.decoding_radius()) sage: y = Chan(c) sage: D.connected_encoder().unencode(c) == D.decode_to_message(y) True

decoding_radius
()¶ Return maximal number of errors that
self
can decodeOUTPUT:
 the number of errors as an integer
EXAMPLES:
sage: F = GF(59) sage: n, k = 40, 12 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: D = codes.decoders.GRSGaoDecoder(C) sage: D.decoding_radius() 14

class
sage.coding.grs.
GRSKeyEquationSyndromeDecoder
(code)¶ Bases:
sage.coding.decoder.Decoder
Decoder for (Generalized) ReedSolomon codes which uses a Key equation decoding based on the syndrome polynomial to correct errors in codewords.
This algorithm uses early terminated extended euclidean algorithm to solve the key equations, as described in [Rot2006], pp. 183195.
INPUT:
code
– The associated code of this decoder.
EXAMPLES:
sage: F = GF(59) sage: n, k = 40, 12 sage: C = codes.GeneralizedReedSolomonCode(F.list()[1:n+1], k) sage: D = codes.decoders.GRSKeyEquationSyndromeDecoder(C) sage: D Key equation decoder for [40, 12, 29] ReedSolomon Code over GF(59)
Actually, we can construct the decoder from
C
directly:sage: D = C.decoder("KeyEquationSyndrome") sage: D Key equation decoder for [40, 12, 29] ReedSolomon Code over GF(59)

decode_to_code
(r)¶ Correct the errors in
r
and returns a codeword.Note
If the code associated to
self
has the same length as its dimension,r
will be returned as is.INPUT:
r
– a vector of the ambient space ofself.code()
OUTPUT:
 a vector of
self.code()
EXAMPLES:
sage: F = GF(59) sage: n, k = 40, 12 sage: C = codes.GeneralizedReedSolomonCode(F.list()[1:n+1], k) sage: D = codes.decoders.GRSKeyEquationSyndromeDecoder(C) sage: c = C.random_element() sage: Chan = channels.StaticErrorRateChannel(C.ambient_space(), D.decoding_radius()) sage: y = Chan(c) sage: c == D.decode_to_code(y) True

decode_to_message
(r)¶ Decode
r
to an element in message space ofself
Note
If the code associated to
self
has the same length as its dimension,r
will be unencoded as is. In that case, ifr
is not a codeword, the output is unspecified.INPUT:
r
– a codeword ofself
OUTPUT:
 a vector of
self
message space
EXAMPLES:
sage: F = GF(59) sage: n, k = 40, 12 sage: C = codes.GeneralizedReedSolomonCode(F.list()[1:n+1], k) sage: D = codes.decoders.GRSKeyEquationSyndromeDecoder(C) sage: c = C.random_element() sage: Chan = channels.StaticErrorRateChannel(C.ambient_space(), D.decoding_radius()) sage: y = Chan(c) sage: D.connected_encoder().unencode(c) == D.decode_to_message(y) True

decoding_radius
()¶ Return maximal number of errors that
self
can decodeOUTPUT:
 the number of errors as an integer
EXAMPLES:
sage: F = GF(59) sage: n, k = 40, 12 sage: C = codes.GeneralizedReedSolomonCode(F.list()[1:n+1], k) sage: D = codes.decoders.GRSKeyEquationSyndromeDecoder(C) sage: D.decoding_radius() 14

class
sage.coding.grs.
GeneralizedReedSolomonCode
(evaluation_points, dimension, column_multipliers=None)¶ Bases:
sage.coding.linear_code.AbstractLinearCode
Representation of a (Generalized) ReedSolomon code.
INPUT:
evaluation_points
– a list of distinct elements of some finite field \(F\)dimension
– the dimension of the resulting codecolumn_multipliers
– (default:None
) list of nonzero elements of \(F\); all column multipliers are set to 1 if default value is kept
EXAMPLES:
Often, one constructs a ReedSolomon code by taking all nonzero elements of the field as evaluation points, and specifying no column multipliers (see also
ReedSolomonCode()
for constructing classical ReedSolomon codes directly):sage: F = GF(7) sage: evalpts = [F(i) for i in range(1,7)] sage: C = codes.GeneralizedReedSolomonCode(evalpts, 3) sage: C [6, 3, 4] ReedSolomon Code over GF(7)
More generally, the following is a ReedSolomon code where the evaluation points are a subset of the field and includes zero:
sage: F = GF(59) sage: n, k = 40, 12 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: C [40, 12, 29] ReedSolomon Code over GF(59)
It is also possible to specify the column multipliers:
sage: F = GF(59) sage: n, k = 40, 12 sage: colmults = F.list()[1:n+1] sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k, colmults) sage: C [40, 12, 29] Generalized ReedSolomon Code over GF(59)

column_multipliers
()¶ Return the vector of column multipliers of
self
.EXAMPLES:
sage: F = GF(11) sage: n, k = 10, 5 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: C.column_multipliers() (1, 1, 1, 1, 1, 1, 1, 1, 1, 1)

covering_radius
()¶ Return the covering radius of
self
.The covering radius of a linear code \(C\) is the smallest number \(r\) s.t. any element of the ambient space of \(C\) is at most at distance \(r\) to \(C\).
As GRS codes are Maximum Distance Separable codes (MDS), their covering radius is always \(d1\), where \(d\) is the minimum distance. This is opposed to random linear codes where the covering radius is computationally hard to determine.
EXAMPLES:
sage: F = GF(2^8, 'a') sage: n, k = 256, 100 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: C.covering_radius() 156

decode_to_message
(r)¶ Decode
r
to an element in message space ofself
.Note
If the code associated to
self
has the same length as its dimension,r
will be unencoded as is. In that case, ifr
is not a codeword, the output is unspecified.INPUT:
r
– a codeword ofself
OUTPUT:
 a vector of
self
message space
EXAMPLES:
sage: F = GF(11) sage: n, k = 10, 5 sage: C = codes.GeneralizedReedSolomonCode(F.list()[1:n+1], k) sage: r = vector(F, (8, 2, 6, 10, 6, 10, 7, 6, 7, 2)) sage: C.decode_to_message(r) (3, 6, 6, 3, 1)

dual_code
()¶ Return the dual code of
self
, which is also a GRS code.EXAMPLES:
sage: F = GF(59) sage: colmults = [ F.random_element() for i in range(40) ] sage: C = codes.GeneralizedReedSolomonCode(F.list()[:40], 12, colmults) sage: Cd = C.dual_code(); Cd [40, 28, 13] Generalized ReedSolomon Code over GF(59)
The dual code of the dual code is the original code:
sage: C == Cd.dual_code() True

evaluation_points
()¶ Return the vector of field elements used for the polynomial evaluations.
EXAMPLES:
sage: F = GF(11) sage: n, k = 10, 5 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: C.evaluation_points() (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)

is_generalized
()¶ Return whether
self
is a Generalized ReedSolomon code or a regular ReedSolomon code.self
is a Generalized ReedSolomon code if its column multipliers are not all 1.EXAMPLES:
sage: F = GF(11) sage: n, k = 10, 5 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: C.column_multipliers() (1, 1, 1, 1, 1, 1, 1, 1, 1, 1) sage: C.is_generalized() False sage: colmults = [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 1] sage: C2 = codes.GeneralizedReedSolomonCode(F.list()[:n], k, colmults) sage: C2.is_generalized() True

minimum_distance
()¶ Return the minimum distance between any two words in
self
.Since a GRS code is always MaximumDistanceSeparable (MDS), this returns
C.length()  C.dimension() + 1
.EXAMPLES:
sage: F = GF(59) sage: n, k = 40, 12 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: C.minimum_distance() 29

multipliers_product
()¶ Return the componentwise product of the column multipliers of
self
with the column multipliers of the dual GRS code.This is a simple Cramer’s rulelike expression on the evaluation points of
self
. Recall that the column multipliers of the dual GRS code are also the column multipliers of the parity check matrix ofself
.EXAMPLES:
sage: F = GF(11) sage: n, k = 10, 5 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: C.multipliers_product() [10, 9, 8, 7, 6, 5, 4, 3, 2, 1]

parity_check_matrix
()¶ Return the parity check matrix of
self
.EXAMPLES:
sage: F = GF(11) sage: n, k = 10, 5 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: C.parity_check_matrix() [10 9 8 7 6 5 4 3 2 1] [ 0 9 5 10 2 3 2 10 5 9] [ 0 9 10 8 8 4 1 4 7 4] [ 0 9 9 2 10 9 6 6 1 3] [ 0 9 7 6 7 1 3 9 8 5]

parity_column_multipliers
()¶ Return the list of column multipliers of the parity check matrix of
self
. They are also column multipliers of the generator matrix for the dual GRS code ofself
.EXAMPLES:
sage: F = GF(11) sage: n, k = 10, 5 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: C.parity_column_multipliers() [10, 9, 8, 7, 6, 5, 4, 3, 2, 1]

weight_distribution
()¶ Return the list whose \(i\)‘th entry is the number of words of weight \(i\) in
self
.Computing the weight distribution for a GRS code is very fast. Note that for random linear codes, it is computationally hard.
EXAMPLES:
sage: F = GF(11) sage: n, k = 10, 5 sage: C = codes.GeneralizedReedSolomonCode(F.list()[:n], k) sage: C.weight_distribution() [1, 0, 0, 0, 0, 0, 2100, 6000, 29250, 61500, 62200]

sage.coding.grs.
ReedSolomonCode
(base_field, length, dimension, primitive_root=None)¶ Construct a classical ReedSolomon code.
A classical \([n,k]\) ReedSolomon code over \(GF(q)\) with \(1 \le k \le n\) and \(n  (q1)\) is a ReedSolomon code whose evaluation points are the consecutive powers of a primitive \(n\)‘th root of unity \(\alpha\), i.e. \(\alpha_i = \alpha^{i1}\), where \(\alpha_1, \ldots, \alpha_n\) are the evaluation points. A classical ReedSolomon codes has all column multipliers equal \(1\).
Classical ReedSolomon codes are cyclic, unlike most Generalized ReedSolomon codes.
Use
GeneralizedReedSolomonCode
if you instead wish to construct nonclassical ReedSolomon and Generalized ReedSolomon codes.INPUT:
base_field
– the finite field for which to build the classical ReedSolomon code.length
– the length of the classical ReedSolomon code. Must divide \(q1\) where \(q\) is the cardinality ofbase_field
.dimension
– the dimension of the resulting code.primitive_root
– (default:None
) a primitive \(n\)‘th root of unity to use for constructing the classical ReedSolomon code. If not supplied, one will be computed and can be recovered asC.evaluation_points()[1]
where \(C\) is the code returned by this method.
EXAMPLES:
sage: C = codes.ReedSolomonCode(GF(7), 6, 3); C [6, 3, 4] ReedSolomon Code over GF(7)
This code is cyclic as can be seen by coercing it into a cyclic code:
sage: Ccyc = codes.CyclicCode(code=C); Ccyc [6, 3] Cyclic Code over GF(7) sage: Ccyc.generator_polynomial() x^3 + 3*x^2 + x + 6
Another example over an extension field:
sage: C = codes.ReedSolomonCode(GF(64,'a'), 9, 4); C [9, 4, 6] ReedSolomon Code over GF(64)
The primitive \(n\)‘th root of unity can be recovered as the 2nd evaluation point of the code:
sage: alpha = C.evaluation_points()[1]; alpha a^5 + a^4 + a^2 + a
We can also supply a different primitive \(n\)‘th root of unity:
sage: beta = alpha^2; beta a^4 + a sage: beta.multiplicative_order() 9 sage: D = codes.ReedSolomonCode(GF(64), 9, 4, primitive_root=beta); D [9, 4, 6] ReedSolomon Code over GF(64) sage: C == D False