Decoders¶
Representation of an errorcorrection algorithm for a code.
AUTHORS:
 David Joyner (20090201): initial version
 David Lucas (20150629): abstract class version

class
sage.coding.decoder.
Decoder
(code, input_space, connected_encoder_name)¶ Bases:
sage.structure.sage_object.SageObject
Abstract topclass for
Decoder
objects.Every decoder class for linear codes (of any metric) should inherit from this abstract class.
To implement an decoder, you need to:
 inherit from
Decoder
 call
Decoder.__init__
in the subclass constructor. Example:super(SubclassName, self).__init__(code, input_space, connected_encoder_name)
. By doing that, your subclass will have all the parameters described above initialized.  Then, you need to override one of decoding methods, either
decode_to_code()
ordecode_to_message()
. You can also override the optional methoddecoding_radius()
.  By default, comparison of
Decoder
(using methods__eq__
and__ne__
) are by memory reference: if you build the same decoder twice, they will be different. If you need something more clever, override__eq__
and__ne__
in your subclass.  As
Decoder
is not designed to be instantiated, it does not have any representation methods. You should implement_repr_
and_latex_
methods in the subclass.

code
()¶ Return the code for this
Decoder
.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: D = C.decoder() sage: D.code() [7, 4] linear code over GF(2)

connected_encoder
()¶ Return the connected encoder 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: D = C.decoder() sage: D.connected_encoder() Generator matrixbased encoder for [7, 4] linear code over GF(2)

decode_to_code
(r)¶ Correct the errors in
r
and returns a codeword.This is a default implementation which assumes that the method
decode_to_message()
has been implemented, else it returns an exception.INPUT:
r
– a element of the input space ofself
.
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), (1, 1, 0, 0, 1, 1, 0)) sage: word in C True sage: w_err = word + vector(GF(2), (1, 0, 0, 0, 0, 0, 0)) sage: w_err in C False sage: D = C.decoder() sage: D.decode_to_code(w_err) (1, 1, 0, 0, 1, 1, 0)

decode_to_message
(r)¶ Decode
r
to the message space ofconnected_encoder()
.This is a default implementation, which assumes that the method
decode_to_code()
has been implemented, else it returns an exception.INPUT:
r
– a element of the input space ofself
.
OUTPUT:
 a vector of
message_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: word = vector(GF(2), (1, 1, 0, 0, 1, 1, 0)) sage: w_err = word + vector(GF(2), (1, 0, 0, 0, 0, 0, 0)) sage: D = C.decoder() sage: D.decode_to_message(w_err) (0, 1, 1, 0)

classmethod
decoder_type
()¶ Returns the set of types of
self
.This method can be called on both an uninstantiated decoder class, or on an instance of a decoder class.
The types of a decoder are a set of labels commonly associated with decoders which describe the nature and behaviour of the decoding algorithm. It should be considered as an informal descriptor but can be coarsely relied upon for e.g. program logic.
The following are the most common types and a brief definition:
Decoder type Definition alwayssucceed The decoder always returns a closest codeword if the number of errors is up to the decoding radius. boundeddistance Any vector with Hamming distance at most decoding_radius()
to a codeword is decodable to some codeword. Ifmightfail
is also a type, then this is not a guarantee but an expectancy.complete The decoder decodes every word in the ambient space of the code. dynamic Some of the decoder’s types will only be determined at construction time (depends on the parameters). halfminimumdistance The decoder corrects up to half the minimum distance, or a specific lower bound thereof. harddecision The decoder uses no information on which positions are more likely to be in error or not. listdecoder The decoder outputs a list of likely codewords, instead of just a single codeword. mightfail The decoder can fail at decoding even within its usual promises, e.g. bounded distance. notalwaysclosest The decoder does not guarantee to always return a closest codeword. probabilistic The decoder has internal randomness which can affect running time and the decoding result. softdecision As part of the input, the decoder takes reliability information on which positions are more likely to be in error. Such a decoder only works for specific channels. EXAMPLES:
We call it on a class:
sage: codes.decoders.LinearCodeSyndromeDecoder.decoder_type() {'dynamic', 'harddecision'}
We can also call it on a instance of a Decoder class:
sage: G = Matrix(GF(2), [[1, 0, 0, 1], [0, 1, 1, 1]]) sage: C = LinearCode(G) sage: D = C.decoder() sage: D.decoder_type() {'complete', 'harddecision', 'mighterror'}

decoding_radius
(**kwargs)¶ Return the maximal number of errors that
self
is able to correct.This is an abstract method and it should be implemented in subclasses.
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: D = codes.decoders.LinearCodeSyndromeDecoder(C) sage: D.decoding_radius() 1

input_space
()¶ Return the input 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: D = C.decoder() sage: D.input_space() Vector space of dimension 7 over Finite Field of size 2

message_space
()¶ Return the message space of
self
’sconnected_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: D = C.decoder() sage: D.message_space() Vector space of dimension 4 over Finite Field of size 2
 inherit from