Opened 3 years ago
Last modified 4 weeks ago
#20970 needs_review enhancement
Gabidulin Codes
Reported by:  arpitdm  Owned by:  

Priority:  major  Milestone:  sage7.3 
Component:  coding theory  Keywords:  
Cc:  dlucas, jsrn, caruso, ghemes4  Merged in:  
Authors:  Arpit Merchant  Reviewers:  
Report Upstream:  N/A  Work issues:  
Branch:  u/ghemes4/coding/gabidulin (Commits)  Commit:  55128af375b6e4ee5c8becf5f07763153364208c 
Dependencies:  #13215, #21131, #21226  Stopgaps: 
Description
A Linear Gabidulin Code Gab[n,k] over F_{q^{m}} of length n <= m and dimension k <= n is the set of all words, formed by the operator evaluation of a qdegree restricted skew polynomial (with frobenius endomorphism and a finite field) f(x) \in S_{F_{q^{m}}}[x].
i.e. Gab[n,k] = {(f(g_0), f(g_1),..., f(g_{n1})) = f(g): deg(f) < k}
where g_0,...,g_{n1} are fixed elements belonging to F_{q^{m}} and are linearly independent over F_{q}
This ticket proposes a new class for Gabidulin Codes along with encoders and decoders for it.
Change History (34)
comment:1 Changed 3 years ago by
 Branch set to u/arpitdm/gabidulin_codes
comment:2 Changed 3 years ago by
 Commit set to ddaef4138cf29d70428a80539d249461226fbc9d
comment:3 Changed 3 years ago by
Hello,
To answer your question (4.
), for now, yes, I guess you need to have such a method.
Later on, when the abstract class to manage rankmetric codes will be implemented we can
replicate the design used in AbstractLinearCode
, but for now, copypasting generator_matrix
from AbstractLinearCode
and put it in GabidulinCode
is fine.
Some remarks:
 You don't have to use backslashes to jump lines when you're in a parenthesis/square bracket block.
For instance, you can write:
% (self.length(), self.dimension(), \ self.minimum_distance(), self.base_field())
as% (self.length(), self.dimension(), self.minimum_distance(), self.base_field())
 Lines 1934 are more or less copypasted from
relative_finite_field_extension.py
, and are not necessary. Take lines 1920 for instance: it's a carbon copy of lines 116117 ofrelative_finite_field_extension.py
(except you replacedmust be
byhas to be
in the error message)... Which basically means you run the same sanity check twice: first time on lines 1920 and second time line 49 when calling the constructing aRelativeFiniteFieldExtension
. I think you should just remove all those tests and letRelativeFiniteFieldExtension
do its job on sanitizing.
 In the same vein: you copypasted a lot of getter from
relative_finite_field_extension.py
, and I'm not sure it makes sense. The user might be a bit puzzled byabsolute_field_power
which returns something of little relevance wrt. the Gabidulin code. Furthermore, if the user desperately needs to access this value, they can still access it by theRelativeFiniteFieldExtension
returned byrelative_finite_field_extension
(this one we should keep).
 There's some kind of naming conflict:
evaluation_points
andlinearly_independent_elements
are the same (see line 68), but while the user has to writelinearly_independent_elements=stuff
at construction time, they have to writemy_code.evaluation_points()
to access them from the constructed code. I think you should choose one name and stick to it. I vote forevaluation_points
for consistency withgrs.py
.
 Remember to not call methods in for loops except it's absolutely necessary. You do this on lines 9697, 180184 and 212. E.g. line 182, as return value of
p.coefficients()
never changes throughout the iterations, I would store it in a local variable and use this local variable instead.
Otherwise it seems good.
Best,
David
comment:4 Changed 3 years ago by
Noticed the following: In one place, you should be computing sigma^t(g)
for some power t
and some element g
, where sigma
is the Frobenius automorphism. You implement this as pow(sigma(g), t)
, but that's not what it means. Rather, it means sigma( sigma( ... sigma(g)...)
, where there is t
sigmas. In another place, you compute simply g^(q^t)
instead of calling sigma
, which is also bad style; it should again be sigma
called t
times on g
.
comment:5 Changed 3 years ago by
Also, avoid "log" if you can:
s = log(q,p)
should bes = relative_field.degree()
. Similar withsm
.m
should besm / m
. Check if the division goes well.
comment:6 Changed 3 years ago by
 Commit changed from ddaef4138cf29d70428a80539d249461226fbc9d to 40a3d3727bad1de2d68fd84fa4abc13dad759d77
Branch pushed to git repo; I updated commit sha1. Last 10 new commits:
de61ca4  Importing ring_element, integral_domain and principal_ideal_domain were throwing deprecation warnings. Updated the import statements according to Tickets 19167 and 20011.

9d21b54  module that implements class 'Side' with Left and Right as the two instances.

9de7bc1  Merge branch 'u/arpitdm/skew_polynomials' of git://trac.sagemath.org/sage into skew_polynomials

9c526ad  Fixed doctests caused by deprecation warnings, inability to access private variables, AttributeError in accessing of parent, and .

a06c2bf  Merge branch 'u/arpitdm/skew_polynomials' of git://trac.sagemath.org/sage into skew_polynomials

18c7982  Fixed bug with integer coercion

c6183bc  Merge branch 'develop' into temp3

dd5c575  Editing declarations of cython functions so as to make them compatible.

60b7869  Merge branch 'u/arpitdm/skew_polynomials' of git://trac.sagemath.org/sage into gabidulin_codes

40a3d37  Refactored code based on comments 3, 4 and 5. Added support for vector_to_matrix and matrix_to_vector representations of codewords. Added methods for rank_weight of a codeword and rank_distance between two codewords.

comment:7 followup: ↓ 10 Changed 3 years ago by
Oh! I thought I only added the gabidulin.py file to my commit. Is there a way I can remove the skew polynomial files from this ticket?
Also, @jsrn, you mention I computed g^{q}t in the generator matrix of the code. According to Equation 2.27 of the PhD thesis, there is no sigma in the formation of the elements of the generator matrix.
comment:8 Changed 3 years ago by
 Commit changed from 40a3d3727bad1de2d68fd84fa4abc13dad759d77 to 799d8713e4f49ff7301633fe6313271ad2082062
Branch pushed to git repo; I updated commit sha1. New commits:
799d871  Added support for in the PolynomialEvaluationEncoder. This includes methods for fast lagrangetype interpolation of skew polynomials, minimum subspace polynomials and multi point evaluation of skew polynomials.

comment:9 Changed 3 years ago by
unencode_nocheck
method in the Polynomial Evaluation Encoder does not work right now because it calls the right division of one skew polynomial by another which fails because of the integer coercion error in #13215. Apart from that, as far as I am able to check, the four new methods are implemented correctly and they do not cause any errors during build or runtime.
comment:10 in reply to: ↑ 7 Changed 3 years ago by
Replying to arpitdm:
Oh! I thought I only added the gabidulin.py file to my commit. Is there a way I can remove the skew polynomial files from this ticket?
I guess you merged in the current tip of #13215 and that's what is showing up as a long list of commits? That's perfectly ok (#13215 is a dependency of this ticket).
Also, @jsrn, you mention I computed g^{q}t in the generator matrix of the code. According to Equation 2.27 of the PhD thesis, there is no sigma in the formation of the elements of the generator matrix.
Well, due to sigma always being the Frobenius, then sigma^t(g)
is, by definition, g^(q^t)
. WachterZeh just chose to write things using explicit q
powering notation everywhere, instead of writing sigma. But it is more mathematically elegant to write sigma, and it fits better with the rest of the code. (say for instance that we choose to make a parameter that you can construct Gabidulin codes with sigma be a power of Frobenius, e.g. sigma(g) = g^(q^2)
. These are sometimes, somewhat ridiculously, called "Generalized Gabidulin codes").
comment:11 Changed 3 years ago by
 Cc caruso added
comment:12 Changed 3 years ago by
 Commit changed from 799d8713e4f49ff7301633fe6313271ad2082062 to e1bf28f3123ddd4188172199ca490b2b5f2b2519
Branch pushed to git repo; I updated commit sha1. Last 10 new commits:
9064e0e  Fix division bug in interpolation

6618340  Fix dividebytwo potential problem in interpolation

0635ec9  Merge branch '13215_skew_polynomials' into 21131_interpolation_skew_poly

e67411f  merge mvp_mpe ticket

392047b  Merge branch 'u/arpitdm/gabidulin_codes' of git://trac.sagemath.org/sage into gabidulin_codes

d97d9c2  removed commented code

4e3e143  updated encode function

16d2d3d  fixed generator matrix method

3c1261f  changed code constructor method to add linearized polynomial ring as mandatory argument

e1bf28f  added Gao decoder

comment:13 Changed 3 years ago by
Hi Arpit,
Just saw your Gao decoder, and I have a few initial comments:
r_
(perhapsR
is a better name) is an interpolation polynomial: construct it using your newinterpolation_polynomial
method instead of what you're manually doing right now (which is much slower).
 After you have performed the division and checked that
rem
is zero (useif rem.is_zero()
), then you need to check thatquo
indeed corresponds to the message of a codeword close tor
. See how the Gao decoder for ReedSolomon codes does this with a helper function, covering bothdecode_to_message
anddecode_to_codeword
in an efficient way.
 You
_partial_xgcd
would be more readable in a compact form, IMO. Say you callr_current > r_c
andr_previous > r_p
, then you can replace the loop body with:
(quo, r_c), r_p = r_c.right_quo_rem(r_p), r_c u_c, u_p = u_p  q*u_c, u_c v_c, v_p = v_p  q*v_c, v_c
Best, Johan
comment:14 followup: ↓ 15 Changed 3 years ago by
Another comment: The Gabidulin constructor shouldn't take the skew polynomial ring (NOT linearized polynomial ring) as an argument, just like ReedSolomon and ReedMuller codes don't take it a polynomial ring as argument. Indeed: the code itself (being a subvector space of GF(q^m)^n
) is invariant of the skew polynomial ring used for encoding the code from a message space. By this logic, the polynomial encoder objects for ReedSolomon and ReedMuller codes can optionally take a polynomial ring, but if the user doesn't supply one, one will be created automatically.
comment:15 in reply to: ↑ 14 ; followup: ↓ 16 Changed 3 years ago by
Replying to jsrn:
Another comment: The Gabidulin constructor shouldn't take the skew polynomial ring (NOT linearized polynomial ring) as an argument, just like ReedSolomon and ReedMuller codes don't take it a polynomial ring as argument. Indeed: the code itself (being a subvector space of
GF(q^m)^n
) is invariant of the skew polynomial ring used for encoding the code from a message space. By this logic, the polynomial encoder objects for ReedSolomon and ReedMuller codes can optionally take a polynomial ring, but if the user doesn't supply one, one will be created automatically.
The reason I put it there is because although the code is a subspace, the skew polynomial ring takes an additional argument, the base ring automorphism. This is not required in a GRS code. It could be Frobenius or a power of the Frobenius, right? And so, a given polynomial could belong to different rings. In the earlier commits I had assumed that the automorphism is the Frobenius.
comment:16 in reply to: ↑ 15 Changed 3 years ago by
The reason I put it there is because although the code is a subspace, the skew polynomial ring takes an additional argument, the base ring automorphism. This is not required in a GRS code. It could be Frobenius or a power of the Frobenius, right? And so, a given polynomial could belong to different rings. In the earlier commits I had assumed that the automorphism is the Frobenius.
True. The Gabidulin code constructor could therefore take as argument /which/ power of the Frobenius (integer between 1 and m1, both inclusive). That feels more natural for the user, and is much less work to specify.
Best, Johan
comment:17 followup: ↓ 18 Changed 3 years ago by
Actually, thanks to the recent PhD thesis of Robert, Gabidulin codes make sense (and seem to have applications) for extensions of infinite fields (e.g. number fields) as well. Moreover I strongly believe (although I have not checked it carefully) and WechterZeh's algorithms extend readily to the general case.
For this reason, I would say that it makes sense to allow more general morphisms that powers of Frobenius. Concretely I propose to replace the current linearized_polynomial_ring
by twisting_homomorphism
(and possibly make relative_field
optional and let the default be the fixed ring under the twisting homomorphism).
comment:18 in reply to: ↑ 17 Changed 3 years ago by
Replying to caruso:
Actually, thanks to the recent PhD thesis of Robert, Gabidulin codes make sense (and seem to have applications) for extensions of infinite fields (e.g. number fields) as well.
That's true and a good point. However, codes over infinite fields needs to be handled specially in many ways, and it would require special casing and some additional testing to do make sure things work. Also, our current plan is to let GabidulinCode
inherit from AbstractLinearCode
, and that class has many things which I know break over infinite fields.
Moreover I strongly believe (although I have not checked it carefully) and WechterZeh's algorithms extend readily to the general case.
I'm also pretty sure it "just works". But it will have terrible performance in the Euclidean algorithm because of unmitigated coefficient growth.
For this reason, I would say that it makes sense to allow more general morphisms that powers of Frobenius. Concretely I propose to replace the current
linearized_polynomial_ring
bytwisting_homomorphism
(and possibly makerelative_field
optional and let the default be the fixed ring under the twisting homomorphism).
For the sake of getting this GSoC finished in time, I would downvote officially supporting infinite fields. But of course, we can still think about the best interface that will scale well in the future. I think your suggestion wrt. these arguments is good if it is practical in the current Sage. For instance, how would I create a Frobenius automorphism from a relative field to an extension field, and such that the Gabidulin code constructor can query the fixed field? The following works poorly:
sage: q = 3^2 sage: F.<a> = GF(q^2, 'a') sage: sigma = F.hom([a^q]) sage: sigma Ring endomorphism of Finite Field in a of size 3^4 Defn: a > a^3 + a^2 + 2*a sage: sigma.fixed_<tab gives nothing>
Best, Johan
comment:19 Changed 3 years ago by
OK for not supporting infinite fields for now. (And I agree that there are issues with growing of coefficients. By the way, do you know if there exists an analogue of subresultants for skew polynomials?)
To answer your last question:
sage: q = 5^2 sage: k.<a> = GF(q^2) sage: Frob = k.frobenius_endomorphism(2) sage: Frob.fixed_field() (Finite Field in a_fixed of size 5^2, Ring morphism: From: Finite Field in a_fixed of size 5^2 To: Finite Field in a of size 5^4 Defn: a_fixed > 4*a^3 + 4*a^2 + 4*a + 3)
comment:20 Changed 3 years ago by
 Commit changed from e1bf28f3123ddd4188172199ca490b2b5f2b2519 to de37ab8ddf5bf0d18be9f49f73554c16242b48fc
Branch pushed to git repo; I updated commit sha1. New commits:
de37ab8  fixed Gao decoder class

comment:21 followup: ↓ 22 Changed 3 years ago by
@arpitdm: Remember to add a message after you push. Your changes to the decoder looks good. I'm somewhat baffled why you changed GabidulinCode
to inherit from AbstractLinearCode
since we have been discussing for some time now that you should create a class AbstractLinearRankMetricCode
.
@caruso: Thanks for the snippet, I hadn't noticed that but of course it's pretty obvious. I'm not sure, though, that I would find this behaviour the most intuitive for Gabidulin codes, especially if relative_field
was automatically determined from the fixed field of twisting_homomorphism
. Fir instance, an unsuspecting user might well do something like:
sage: F = GF(5^20,'a') sage: evals = [ a^i for i in range(20) ] sage: C = GabidulinCode(evals, k=5) # twisting_homomorphism taken to be absolute Frob. sage: <fun experiments with C.> sage: C2 = GabidulinCode(evals, k=5, twisting_homomorphism=F.frobenius_endomorphism(2)) <BOOM: Some error probably stating that the evaluation points are not linearly independent over the subfield>
A carefully phrased exception at the end would help the user find the bug in an interactive session, but I think this is the kind of surprising behaviour that one keeps on running into every time one uses the functionality.
I would prefer a session looking something like this:
sage: F = GF(5^20,'a') sage: evals = [ a^i for i in range(20) ] # unless specified, subfield = prime field, frobenius power = 1 sage: C = GabidulinCode(evals, k=5) sage: C [20,5,16] Gabidulin code over Finite Field in a of size 5^20 with subfield of size 5 sage: C2 = GabidulinCode(evals, k=5, frobenius_power=2) #subfield is still prime field sage: C2 [20,5,16] Gabidulin code with Frobenius power 2 over Finite Field in a of size 5^20 with subfield of size 5 sage: C3 = GabidulinCode([a^i for i in range(10)], k=5, frobenius_power=2, subfield=GF(5^2,'b')) sage: C3 [10,5,6] Gabidulin code over Finite Field in a of size 5^20 with subfield of size 5^2.
For your other question: I'm not aware of subresultants for skew polynomials, no. I've recently started looking at computations in skew polynomials in the more general setting of derivations, and over infinite fields. All algorithms I know of perform pretty terribly compared to the case without coefficient growth: the usual approaches of homomorphic imaging doesn't seem to work. Such rings seem more widely known as Ore polynomials. We should add a note about this in the doc of #13215.
comment:22 in reply to: ↑ 21 ; followup: ↓ 23 Changed 3 years ago by
Replying to jsrn:
@arpitdm: Remember to add a message after you push. Your changes to the decoder looks good. I'm somewhat baffled why you changed
GabidulinCode
to inherit fromAbstractLinearCode
since we have been discussing for some time now that you should create a classAbstractLinearRankMetricCode
.
I was about to when discussion started on #13215. Anyway, in the _decode_to_code_and_message
, I needed to use the connected_encoder
method which is available in the AbstractLinearCode? class. Since ARMC is not yet opened, I inherited this temporarily from that. I will update to ARMC once it is possible.
comment:23 in reply to: ↑ 22 Changed 3 years ago by
Replying to arpitdm:
I was about to when discussion started on #13215. Anyway, in the
_decode_to_code_and_message
, I needed to use theconnected_encoder
method which is available in the AbstractLinearCode? class. Since ARMC is not yet opened, I inherited this temporarily from that. I will update to ARMC once it is possible.
It's a method on Decoder
 not AbstractLinearCode
. But I can see how it could be practical to temporarily sidestep ARMC until it's there.
comment:24 Changed 3 years ago by
 Commit changed from de37ab8ddf5bf0d18be9f49f73554c16242b48fc to 72784538cec186b562d27f4ba825b5cd60bd332b
Branch pushed to git repo; I updated commit sha1. Last 10 new commits:
b9f8239  merging latest changes

34bebbc  created top level functions for interpolation and fraction field skew polynomial ring. added documentation and indirect doctests.

b523806  changed names of private methods

e435f56  Merge branch 'u/arpitdm/mvp_mpe' of git://trac.sagemath.org/sage into mvp_mpe

d403fae  fixes to documentation

7e4b3a3  added a very basic constructor, a couple of basic getter methods and deleted a couple of methods from ALC class. added some rank distance, rank weight, to_matrix_representation and from_matrix_representation.

0657293  Merge branch 'u/arpitdm/abstract_linear_rank_metric_code_class' of git://trac.sagemath.org/sage into abstract_rank_metric_code

08aef71  merging with updates from other tickets

da374e9  small fix to docs

7278453  added random_element method. refactored some code. added documentation and tests.

comment:25 followup: ↓ 27 Changed 3 years ago by
I've added refactored and scrubbed the code to make it cleaner where ever possible. I've created proper inheritances for the classes, and added some methods. And I've added documentation and tests for nearly every method and class. There are two issues that remain. One is def parity_evaluation_points
where the problem is to find a concrete, nontrivial solution for the equation Ax = 0
where A
is the coefficient matrix. I am not sure how to do that in Sage.
And the second is, finding the right name for the getter method on line 321 (currently it is left as def m
).
comment:26 Changed 3 years ago by
 Dependencies changed from #13215 to #13215, #21088, #21131, #21226
comment:27 in reply to: ↑ 25 ; followup: ↓ 28 Changed 3 years ago by
Replying to arpitdm:
I've added refactored and scrubbed the code to make it cleaner where ever possible. I've created proper inheritances for the classes, and added some methods. And I've added documentation and tests for nearly every method and class.
I've looked through the code in overview, and it looks good! Some comments on the structure:
 Constructor of Gabidulin code: you require both
subfield
and twist map. I've thought a bit more about it, and I think that thesubfield
should always be the fixed field of the twist map: if it is smaller than the fixed field, then the code is not MRD, and if it is larger, then the code is not linear. Therefore, I suggest that both twist map andsubfield
is optional: if twist map is set andsubfield
is not, thensubfield
is the fixed field of the twist map; throw an exception if the twist map does not have thefixed_field
method. Ifsubfield
is given but twist map is not, then default to Frobenius of that field extension. If both twist map andsubfield
is set, then throw an exception if the twist map has a methodfixed_field
and it does not return the givensubfield
.
 You don't have an example using
field_extension
.
 Your
parity_check_matrix
is strange. Why not simplyreturn self.dual_code().generator_matrix()
?
 What's the point of
_vector_space
anddef vector_space
?
 You shouldn't override
generator_matrix
andrandom_element
.
 What's the point of
GabidulinCode.message_space
?
 Doc: There's no such thing a as a qdegree of a skew polynomial. The doc should mention that over finite fields, the skew polynomials are very similar to linearized polynomials (isomorphic when the twist map is powerbyq).
There are two issues that remain. One is
def parity_evaluation_points
where the problem is to find a concrete, nontrivial solution for the equationAx = 0
whereA
is the coefficient matrix. I am not sure how to do that in Sage.
Just proceed exactly as you have done, until you get solution_space
. Then a nonzero element is solution_space.basis()[0]
, assuming that solution_space
is not empty.
And the second is, finding the right name for the getter method on line 321 (currently it is left as
def m
).
That should be a method on AbstractLinearRankMetricCode
. It could be called field_extension_degree
.
Best, Johan
comment:28 in reply to: ↑ 27 ; followup: ↓ 29 Changed 3 years ago by
Replying to jsrn:
 Constructor of Gabidulin code: you require both
subfield
and twist map. I've thought a bit more about it, and I think that thesubfield
should always be the fixed field of the twist map: if it is smaller than the fixed field, then the code is not MRD, and if it is larger, then the code is not linear. Therefore, I suggest that both twist map andsubfield
is optional: if twist map is set andsubfield
is not, thensubfield
is the fixed field of the twist map; throw an exception if the twist map does not have thefixed_field
method. Ifsubfield
is given but twist map is not, then default to Frobenius of that field extension. If both twist map andsubfield
is set, then throw an exception if the twist map has a methodfixed_field
and it does not return the givensubfield
.
What do I do if both subfield
and twist_map
are not given?
Also, should I remove the dependency on #21088?
comment:29 in reply to: ↑ 28 Changed 3 years ago by
Replying to arpitdm:
What do I do if both
subfield
andtwist_map
are not given?
Then take twist_map
to be absolute frobenius, i.e. frobenius wrt. the prime field, and subfield
to be the prime field.
Also, should I remove the dependency on #21088?
Oh, yeah I guess you can, since finite fields are already formally supported without #21088.
comment:30 Changed 5 weeks ago by
 Cc ghemes4 added
comment:31 Changed 4 weeks ago by
 Dependencies changed from #13215, #21088, #21131, #21226 to #13215, #21131, #21226
comment:32 Changed 4 weeks ago by
 Branch changed from u/arpitdm/gabidulin_codes to u/ghemes4/coding/gabidulin
 Commit 72784538cec186b562d27f4ba825b5cd60bd332b deleted
comment:33 Changed 4 weeks ago by
 Commit set to 55128af375b6e4ee5c8becf5f07763153364208c
Branch pushed to git repo; I updated commit sha1. Last 10 new commits:
1e32a0c  Super method of LinearRankMetricCode includes basis.

09a006e  Merge 21226

62e0ed2  Merge branch 'develop' of git://trac.sagemath.org/sage into gabidulin

3917048  Merge branch 'develop' of git://trac.sagemath.org/sage into rank_metric

01d9a3d  Merge branch 'develop' of git://trac.sagemath.org/sage into t/28350/abstract_linear_code_no_metric_class

226ffbf  Added no metric to coding documentation index. Moved zero method from AbstractLinearCode. Changed base_field check.

bd31704  Merge branch 'u/ghemes4/coding/no_metric' of git://trac.sagemath.org/sage into rank_metric

0a115d0  Removed zero method. Added field extension method.

9d74474  Merge branch 'u/ghemes4/coding/linear_rank_metric' of git://trac.sagemath.org/sage into gabidulin

55128af  Initial completed, working version. No Metric changes. Documentation and doctests.

comment:34 Changed 4 weeks ago by
 Status changed from new to needs_review
I implemented all of the changes from Johan's last comment.
I fixed all the bugs, added documentation and doctests. Changed the code to reflect all the structural changes in the coding module. I tested the GaoDecoder
method and it seems to be working.
A few points:
solve_right
returns the trivial (allzeros) solution.def generator_matrix
method within the code class itself?New commits:
Initial construction of the Gabidulin Code class. PolynomialEvaluationEncoder and GeneratorMatrixEncoder also included.