Encoders¶
Representation of a bijection between a message space and a code.
AUTHORS:
 David Lucas (2015): initial version

class
sage.coding.encoder.
Encoder
(code)¶ Bases:
sage.structure.sage_object.SageObject
Abstract topclass for
Encoder
objects.Every encoder class for linear codes (of any metric) should inherit from this abstract class.
To implement an encoder, you need to:
 inherit from
Encoder
,  call
Encoder.__init__
in the subclass constructor. Example:super(SubclassName, self).__init__(code)
. By doing that, your subclass will have itscode
parameter initialized.  Then, if the message space is a vector space, default implementations of
encode()
andunencode_nocheck()
methods are provided. These implementations rely ongenerator_matrix()
which you need to override to use the default implementations.  If the message space is not of the form \(F^k\), where \(F\) is a finite field,
you cannot have a generator matrix.
In that case, you need to override
encode()
,unencode_nocheck()
andmessage_space()
.  By default, comparison of
Encoder
(using methods__eq__
and__ne__
) are by memory reference: if you build the same encoder twice, they will be different. If you need something more clever, override__eq__
and__ne__
in your subclass.  As
Encoder
is not designed to be instantiated, it does not have any representation methods. You should implement_repr_
and_latex_
methods in the subclass.
REFERENCES:

code
()¶ Returns the code for this
Encoder
.EXAMPLES:
sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]]) sage: C = LinearCode(G) sage: E = C.encoder() sage: E.code() == C True

encode
(word)¶ Transforms an element of the message space into a codeword.
This is a default implementation which assumes that the message space of the encoder is \(F^{k}\), where \(F\) is
sage.coding.linear_code.AbstractLinearCode.base_field()
and \(k\) issage.coding.linear_code.AbstractLinearCode.dimension()
. If this is not the case, this method should be overwritten by the subclass.Note
encode()
might be a partial function overself
’smessage_space()
. One should use the exceptionEncodingError
to catch attempts to encode words that are outside of the message space.One can use the following shortcut to encode a word with an encoder
E
:E(word)
INPUT:
word
– a vector of the message space of theself
.
OUTPUT:
 a vector of
code()
.
EXAMPLES:
sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]]) sage: C = LinearCode(G) sage: word = vector(GF(2), (0, 1, 1, 0)) sage: E = codes.encoders.LinearCodeGeneratorMatrixEncoder(C) sage: E.encode(word) (1, 1, 0, 0, 1, 1, 0)
If
word
is not in the message space ofself
, it will return an exception:sage: word = random_vector(GF(7), 4) sage: E.encode(word) Traceback (most recent call last): ... ArithmeticError: reduction modulo 2 not defined

generator_matrix
()¶ Returns a generator matrix of the associated code of
self
.This is an abstract method and it should be implemented separately. Reimplementing this for each subclass of
Encoder
is not mandatory (as a generator matrix only makes sense when the message space is of the \(F^k\), where \(F\) is the base field ofcode()
.)EXAMPLES:
sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]]) sage: C = LinearCode(G) sage: E = C.encoder() sage: E.generator_matrix() [1 1 1 0 0 0 0] [1 0 0 1 1 0 0] [0 1 0 1 0 1 0] [1 1 0 1 0 0 1]

message_space
()¶ Returns the ambient space of allowed input to
encode()
. Note thatencode()
is possibly a partial function over the ambient space.EXAMPLES:
sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]]) sage: C = LinearCode(G) sage: E = C.encoder() sage: E.message_space() Vector space of dimension 4 over Finite Field of size 2

unencode
(c, nocheck=False)¶ Return the message corresponding to the codeword
c
.This is the inverse of
encode()
.INPUT:
c
– a codeword ofcode()
.nocheck
– (default:False
) checks ifc
is incode()
. You might set this toTrue
to disable the check for saving computation. Note that ifc
is not inself()
andnocheck = True
, then the output ofunencode()
is not defined (except that it will be in the message space ofself
).
OUTPUT:
 an element of the message space of
self
EXAMPLES:
sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]]) sage: C = LinearCode(G) sage: c = vector(GF(2), (1, 1, 0, 0, 1, 1, 0)) sage: c in C True sage: E = codes.encoders.LinearCodeGeneratorMatrixEncoder(C) sage: E.unencode(c) (0, 1, 1, 0)

unencode_nocheck
(c)¶ Returns the message corresponding to
c
.When
c
is not a codeword, the output is unspecified.AUTHORS:
This function is taken from codinglib [Nie]INPUT:
c
– a codeword ofcode()
.
OUTPUT:
 an element of the message space of
self
.
EXAMPLES:
sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]]) sage: C = LinearCode(G) sage: c = vector(GF(2), (1, 1, 0, 0, 1, 1, 0)) sage: c in C True sage: E = codes.encoders.LinearCodeGeneratorMatrixEncoder(C) sage: E.unencode_nocheck(c) (0, 1, 1, 0)
Taking a vector that does not belong to
C
will not raise an error but probably just give a nonsensical result:sage: c = vector(GF(2), (1, 1, 0, 0, 1, 1, 1)) sage: c in C False sage: E = codes.encoders.LinearCodeGeneratorMatrixEncoder(C) sage: E.unencode_nocheck(c) (0, 1, 1, 0) sage: m = vector(GF(2), (0, 1, 1, 0)) sage: c1 = E.encode(m) sage: c == c1 False
 inherit from