Channels#

Given an input space and an output space, a channel takes element from the input space (the message) and transforms it into an element of the output space (the transmitted message).

In Sage, Channels simulate error-prone transmission over communication channels, and we borrow the nomenclature from communication theory, such as “transmission” and “positions” as the elements of transmitted vectors. Transmission can be achieved with two methods:

• Channel.transmit(). Considering a channel Chan and a message msg, transmitting msg with Chan can be done this way:

Chan.transmit(msg)

It can also be written in a more convenient way:

Chan(msg)

• transmit_unsafe(). This does the exact same thing as transmit() except that it does not check if msg belongs to the input space of Chan:

Chan.transmit_unsafe(msg)

This is useful in e.g. an inner-loop of a long simulation as a lighter-weight alternative to Channel.transmit().

This file contains the following elements:

• Channel, the abstract class for Channels

• StaticErrorRateChannel, which creates a specific number of errors in each transmitted message

• ErrorErasureChannel, which creates a specific number of errors and a specific number of erasures in each transmitted message

class sage.coding.channel.Channel(input_space, output_space)#

Abstract top-class for Channel objects.

All channel objects must inherit from this class. To implement a channel subclass, one should do the following:

While not being mandatory, it might be useful to reimplement representation methods (_repr_ and _latex_).

This abstract class provides the following parameters:

• input_space – the space of the words to transmit

• output_space – the space of the transmitted words

input_space()#

Return the input space of self.

EXAMPLES:

sage: n_err = 2
sage: Chan = channels.StaticErrorRateChannel(GF(59)^6, n_err)
sage: Chan.input_space()
Vector space of dimension 6 over Finite Field of size 59
output_space()#

Return the output space of self.

EXAMPLES:

sage: n_err = 2
sage: Chan = channels.StaticErrorRateChannel(GF(59)^6, n_err)
sage: Chan.output_space()
Vector space of dimension 6 over Finite Field of size 59
transmit(message)#

Return message, modified accordingly with the algorithm of the channel it was transmitted through.

Checks if message belongs to the input space, and returns an exception if not. Note that message itself is never modified by the channel.

INPUT:

• message – a vector

OUTPUT:

• a vector of the output space of self

EXAMPLES:

sage: F = GF(59)^6
sage: n_err = 2
sage: Chan = channels.StaticErrorRateChannel(F, n_err)
sage: msg = F((4, 8, 15, 16, 23, 42))
sage: set_random_seed(10)
sage: Chan.transmit(msg)
(4, 8, 4, 16, 23, 53)

We can check that the input msg is not modified:

sage: msg
(4, 8, 15, 16, 23, 42)

If we transmit a vector which is not in the input space of self:

sage: n_err = 2
sage: Chan = channels.StaticErrorRateChannel(GF(59)^6, n_err)
sage: msg = (4, 8, 15, 16, 23, 42)
sage: Chan.transmit(msg)
Traceback (most recent call last):
...
TypeError: Message must be an element of the input space for the given channel

Note

One can also call directly Chan(message), which does the same as Chan.transmit(message)

transmit_unsafe(message)#

Return message, modified accordingly with the algorithm of the channel it was transmitted through.

This method does not check if message belongs to the input space ofself.

This is an abstract method which should be reimplemented in all the subclasses of Channel.

EXAMPLES:

sage: n_err = 2
sage: Chan = channels.StaticErrorRateChannel(GF(59)^6, n_err)
sage: v = Chan.input_space().random_element()
sage: Chan.transmit_unsafe(v)  # random
(1, 33, 46, 18, 20, 49)
class sage.coding.channel.ErrorErasureChannel(space, number_errors, number_erasures)#

Channel which adds errors and erases several positions in any message it transmits.

The output space of this channel is a Cartesian product between its input space and a VectorSpace of the same dimension over GF(2)

INPUT:

• space – the input and output space

• number_errors – the number of errors created in each transmitted message. It can be either an integer of a tuple. If an tuple is passed as an argument, the number of errors will be a random integer between the two bounds of this tuple.

• number_erasures – the number of erasures created in each transmitted message. It can be either an integer of a tuple. If an tuple is passed as an argument, the number of erasures will be a random integer between the two bounds of this tuple.

EXAMPLES:

We construct a ErrorErasureChannel which adds 2 errors and 2 erasures to any transmitted message:

sage: n_err, n_era = 2, 2
sage: Chan = channels.ErrorErasureChannel(GF(59)^40, n_err, n_era)
sage: Chan
Error-and-erasure channel creating 2 errors and 2 erasures
of input space Vector space of dimension 40 over Finite Field of size 59
and output space The Cartesian product of (Vector space of dimension 40
over Finite Field of size 59, Vector space of dimension 40 over Finite Field of size 2)

We can also pass the number of errors and erasures as a couple of integers:

sage: n_err, n_era = (1, 10), (1, 10)
sage: Chan = channels.ErrorErasureChannel(GF(59)^40, n_err, n_era)
sage: Chan
Error-and-erasure channel creating between 1 and 10 errors and
between 1 and 10 erasures of input space Vector space of dimension 40
over Finite Field of size 59 and output space The Cartesian product of
(Vector space of dimension 40 over Finite Field of size 59,
Vector space of dimension 40 over Finite Field of size 2)
number_erasures()#

Returns the number of erasures created by self.

EXAMPLES:

sage: n_err, n_era = 0, 3
sage: Chan = channels.ErrorErasureChannel(GF(59)^6, n_err, n_era)
sage: Chan.number_erasures()
(3, 3)
number_errors()#

Returns the number of errors created by self.

EXAMPLES:

sage: n_err, n_era = 3, 0
sage: Chan = channels.ErrorErasureChannel(GF(59)^6, n_err, n_era)
sage: Chan.number_errors()
(3, 3)
transmit_unsafe(message)#

Returns message with as many errors as self._number_errors in it, and as many erasures as self._number_erasures in it.

If self._number_errors was passed as an tuple for the number of errors, it will pick a random integer between the bounds of the tuple and use it as the number of errors. It does the same with self._number_erasures.

All erased positions are set to 0 in the transmitted message. It is guaranteed that the erasures and the errors will never overlap: the received message will always contains exactly as many errors and erasures as expected.

This method does not check if message belongs to the input space ofself.

INPUT:

• message – a vector

OUTPUT:

• a couple of vectors, namely:

• the transmitted message, which is message with erroneous and erased positions

• the erasure vector, which contains 1 at the erased positions of the transmitted message , 0 elsewhere.

EXAMPLES:

sage: F = GF(59)^11
sage: n_err, n_era = 2, 2
sage: Chan = channels.ErrorErasureChannel(F, n_err, n_era)
sage: msg = F((3, 14, 15, 9, 26, 53, 58, 9, 7, 9, 3))
sage: set_random_seed(10)
sage: Chan.transmit_unsafe(msg)
((31, 0, 15, 9, 38, 53, 58, 9, 0, 9, 3), (0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0))
class sage.coding.channel.QarySymmetricChannel(space, epsilon)#

The q-ary symmetric, memoryless communication channel.

Given an alphabet $$\Sigma$$ with $$|\Sigma| = q$$ and an error probability $$\epsilon$$, a q-ary symmetric channel sends an element of $$\Sigma$$ into the same element with probability $$1 - \epsilon$$, and any one of the other $$q - 1$$ elements with probability $$\frac{\epsilon}{q - 1}$$. This implementation operates over vectors in $$\Sigma^n$$, and “transmits” each element of the vector independently in the above manner.

Though $$\Sigma$$ is usually taken to be a finite field, this implementation allows any structure for which Sage can represent $$\Sigma^n$$ and for which $$\Sigma$$ has a random_element() method. However, beware that if $$\Sigma$$ is infinite, errors will not be uniformly distributed (since random_element() does not draw uniformly at random).

The input space and the output space of this channel are the same: $$\Sigma^n$$.

INPUT:

• space – the input and output space of the channel. It has to be $$GF(q)^n$$ for some finite field $$GF(q)$$.

• epsilon – the transmission error probability of the individual elements.

EXAMPLES:

We construct a QarySymmetricChannel which corrupts 30% of all transmitted symbols:

sage: epsilon = 0.3
sage: Chan = channels.QarySymmetricChannel(GF(59)^50, epsilon)
sage: Chan
q-ary symmetric channel with error probability 0.300000000000000,
of input and output space Vector space of dimension 50 over Finite Field of size 59
error_probability()#

Returns the error probability of a single symbol transmission of self.

EXAMPLES:

sage: epsilon = 0.3
sage: Chan = channels.QarySymmetricChannel(GF(59)^50, epsilon)
sage: Chan.error_probability()
0.300000000000000
probability_of_at_most_t_errors(t)#

Returns the probability self has to return at most t errors.

INPUT:

• t – an integer

EXAMPLES:

sage: epsilon = 0.3
sage: Chan = channels.QarySymmetricChannel(GF(59)^50, epsilon)
sage: Chan.probability_of_at_most_t_errors(20)
0.952236164579467
probability_of_exactly_t_errors(t)#

Returns the probability self has to return exactly t errors.

INPUT:

• t – an integer

EXAMPLES:

sage: epsilon = 0.3
sage: Chan = channels.QarySymmetricChannel(GF(59)^50, epsilon)
sage: Chan.probability_of_exactly_t_errors(15)
0.122346861835401
transmit_unsafe(message)#

Returns message where each of the symbols has been changed to another from the alphabet with probability error_probability().

This method does not check if message belongs to the input space ofself.

INPUT:

• message – a vector

EXAMPLES:

sage: F = GF(59)^11
sage: epsilon = 0.3
sage: Chan = channels.QarySymmetricChannel(F, epsilon)
sage: msg = F((3, 14, 15, 9, 26, 53, 58, 9, 7, 9, 3))
sage: set_random_seed(10)
sage: Chan.transmit_unsafe(msg)
(3, 14, 15, 53, 12, 53, 58, 9, 55, 9, 3)
class sage.coding.channel.StaticErrorRateChannel(space, number_errors)#

Channel which adds a static number of errors to each message it transmits.

The input space and the output space of this channel are the same.

INPUT:

• space – the space of both input and output

• number_errors – the number of errors added to each transmitted message It can be either an integer of a tuple. If a tuple is passed as argument, the number of errors will be a random integer between the two bounds of the tuple.

EXAMPLES:

We construct a StaticErrorRateChannel which adds 2 errors to any transmitted message:

sage: n_err = 2
sage: Chan = channels.StaticErrorRateChannel(GF(59)^40, n_err)
sage: Chan
Static error rate channel creating 2 errors, of input and output space
Vector space of dimension 40 over Finite Field of size 59

We can also pass a tuple for the number of errors:

sage: n_err = (1, 10)
sage: Chan = channels.StaticErrorRateChannel(GF(59)^40, n_err)
sage: Chan
Static error rate channel creating between 1 and 10 errors,
of input and output space Vector space of dimension 40 over Finite Field of size 59
number_errors()#

Returns the number of errors created by self.

EXAMPLES:

sage: n_err = 3
sage: Chan = channels.StaticErrorRateChannel(GF(59)^6, n_err)
sage: Chan.number_errors()
(3, 3)
transmit_unsafe(message)#

Returns message with as many errors as self._number_errors in it.

If self._number_errors was passed as a tuple for the number of errors, it will pick a random integer between the bounds of the tuple and use it as the number of errors.

This method does not check if message belongs to the input space ofself.

INPUT:

• message – a vector

OUTPUT:

• a vector of the output space

EXAMPLES:

sage: F = GF(59)^6
sage: n_err = 2
sage: Chan = channels.StaticErrorRateChannel(F, n_err)
sage: msg = F((4, 8, 15, 16, 23, 42))
sage: set_random_seed(10)
sage: Chan.transmit_unsafe(msg)
(4, 8, 4, 16, 23, 53)

This checks that trac ticket #19863 is fixed:

sage: V = VectorSpace(GF(2), 1000)
sage: Chan = channels.StaticErrorRateChannel(V, 367)
sage: c = V.random_element()
sage: (c - Chan(c)).hamming_weight()
367
sage.coding.channel.format_interval(t)#

Return a formatted string representation of t.

This method should be called by any representation function in Channel classes.

Note

This is a helper function, which should only be used when implementing new channels.

INPUT:

• t – a list or a tuple

OUTPUT:

• a string

sage.coding.channel.random_error_vector(n, F, error_positions)#

Return a vector of length n over F filled with random non-zero coefficients at the positions given by error_positions.

Note

This is a helper function, which should only be used when implementing new channels.

INPUT:

• n – the length of the vector

• F – the field over which the vector is defined

• error_positions – the non-zero positions of the vector

OUTPUT:

• a vector of F

AUTHORS:

This function is taken from codinglib (https://bitbucket.org/jsrn/codinglib/) and was written by Johan Nielsen.

EXAMPLES:

sage: from sage.coding.channel import random_error_vector
sage: random_error_vector(5, GF(2), [1,3])
(0, 1, 0, 1, 0)