# Information-set decoding for linear codes¶

Information-set decoding is a probabilistic decoding strategy that essentially tries to guess $$k$$ correct positions in the received word, where $$k$$ is the dimension of the code. A codeword agreeing with the received word on the guessed position can easily be computed, and their difference is one possible error vector. A “correct” guess is assumed when this error vector has low Hamming weight.

This simple algorithm is not very efficient in itself, but there are numerous refinements to the strategy that make it very capable over rather large codes. Still, the decoding algorithm is exponential in dimension of the code and the log of the field size.

The ISD strategy requires choosing how many errors is deemed acceptable. One choice could be $$d/2$$, where $$d$$ is the minimum distance of the code, but sometimes $$d$$ is not known, or sometimes more errors are expected. If one chooses anything above $$d/2$$, the algorithm does not guarantee to return a nearest codeword.

AUTHORS:

• David Lucas, Johan Rosenkilde, Yann Laigle-Chapuy (2016-02, 2017-06): initial version

class sage.coding.information_set_decoder.InformationSetAlgorithm(code, decoding_interval, algorithm_name, parameters=None)

Abstract class for algorithms for sage.coding.information_set_decoder.LinearCodeInformationSetDecoder.

To sub-class this class, override decode and calibrate, and call the super constructor from __init__.

INPUT:

• code – A linear code for which to decode.

• number_errors – an integer, the maximal number of errors to accept as correct decoding. An interval can also be specified by giving a pair of integers, where both end values are taken to be in the interval.

• algorithm_name – A name for the specific ISD algorithm used (used for printing).

• parameters – (optional) A dictionary for setting the parameters of this ISD algorithm. Note that sanity checking this dictionary for the individual sub-classes should be done in the sub-class constructor.

EXAMPLES:

sage: from sage.coding.information_set_decoder import LeeBrickellISDAlgorithm
sage: LeeBrickellISDAlgorithm(codes.GolayCode(GF(2)), (0,4))
ISD Algorithm (Lee-Brickell) for [24, 12, 8] Extended Golay code over GF(2) decoding up to 4 errors


A minimal working example of how to sub-class:

sage: from sage.coding.information_set_decoder import InformationSetAlgorithm
sage: from sage.coding.decoder import DecodingError
sage: class MinimalISD(InformationSetAlgorithm):
....:     def __init__(self, code, decoding_interval):
....:         super(MinimalISD, self).__init__(code, decoding_interval, "MinimalISD")
....:     def calibrate(self):
....:         self._parameters = { } # calibrate parameters here
....:         self._time_estimate = 10.0  # calibrated time estimate
....:     def decode(self, r):
....:         # decoding algorithm here
....:         raise DecodingError("I failed")
sage: MinimalISD(codes.GolayCode(GF(2)), (0,4))
ISD Algorithm (MinimalISD) for [24, 12, 8] Extended Golay code over GF(2) decoding up to 4 errors

calibrate()

Uses test computations to estimate optimal values for any parameters this ISD algorithm may take.

Must be overridden by sub-classes.

If self._parameters_specified is False, this method shall set self._parameters to the best parameters estimated. It shall always set self._time_estimate to the time estimate of using self._parameters.

EXAMPLES:

sage: from sage.coding.information_set_decoder import LeeBrickellISDAlgorithm
sage: C = codes.GolayCode(GF(2))
sage: A = LeeBrickellISDAlgorithm(C, (0,3))
sage: A.calibrate()
sage: A.parameters() #random
{'search_size': 1}

code()

Return the code associated to this ISD algorithm.

EXAMPLES:

sage: from sage.coding.information_set_decoder import LeeBrickellISDAlgorithm
sage: C = codes.GolayCode(GF(2))
sage: A = LeeBrickellISDAlgorithm(C, (0,3))
sage: A.code()
[24, 12, 8] Extended Golay code over GF(2)

decode(r)

Decode a received word using this ISD decoding algorithm.

Must be overridden by sub-classes.

EXAMPLES:

sage: M = matrix(GF(2), [[1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0],\
[0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1],\
[0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0],\
[0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1],\
[0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1]])
sage: C = codes.LinearCode(M)
sage: from sage.coding.information_set_decoder import LeeBrickellISDAlgorithm
sage: A = LeeBrickellISDAlgorithm(C, (2,2))
sage: r = vector(GF(2), [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0])
sage: A.decode(r)
(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)

decoding_interval()

A pair of integers specifying the interval of number of errors this ISD algorithm will attempt to correct.

The interval includes both end values.

EXAMPLES:

sage: C = codes.GolayCode(GF(2))
sage: from sage.coding.information_set_decoder import LeeBrickellISDAlgorithm
sage: A = LeeBrickellISDAlgorithm(C, (0,2))
sage: A.decoding_interval()
(0, 2)

name()

Return the name of this ISD algorithm.

EXAMPLES:

sage: C = codes.GolayCode(GF(2))
sage: from sage.coding.information_set_decoder import LeeBrickellISDAlgorithm
sage: A = LeeBrickellISDAlgorithm(C, (0,2))
sage: A.name()
'Lee-Brickell'

parameters()

Return any parameters this ISD algorithm uses.

If the parameters have not already been set, efficient values will first be calibrated and returned.

EXAMPLES:

sage: C = codes.GolayCode(GF(2))
sage: from sage.coding.information_set_decoder import LeeBrickellISDAlgorithm
sage: A = LeeBrickellISDAlgorithm(C, (0,4), search_size=3)
sage: A.parameters()
{'search_size': 3}


If not set, calibration will determine a sensible value:

sage: A = LeeBrickellISDAlgorithm(C, (0,4))
sage: A.parameters() #random
{'search_size': 1}

time_estimate()

Estimate for how long this ISD algorithm takes to perform a single decoding.

The estimate is for a received word whose number of errors is within the decoding interval of this ISD algorithm.

EXAMPLES:

sage: C = codes.GolayCode(GF(2))
sage: from sage.coding.information_set_decoder import LeeBrickellISDAlgorithm
sage: A = LeeBrickellISDAlgorithm(C, (0,2))
sage: A.time_estimate() #random
0.0008162108571427874

class sage.coding.information_set_decoder.LeeBrickellISDAlgorithm(code, decoding_interval, search_size=None)

The Lee-Brickell algorithm for information-set decoding.

For a description of the information-set decoding paradigm (ISD), see sage.coding.information_set_decoder.LinearCodeInformationSetDecoder.

This implements the Lee-Brickell variant of ISD, see [LB1988] for the original binary case, and [Pet2010] for the $$q$$-ary extension.

Let $$C$$ be a $$[n, k]$$-linear code over $$GF(q)$$, and let $$r \in GF(q)^{n}$$ be a received word in a transmission. We seek the codeword whose Hamming distance from $$r$$ is minimal. Let $$p$$ and $$w$$ be integers, such that $$0\leq p\leq w$$, Let $$G$$ be a generator matrix of $$C$$, and for any set of indices $$I$$, we write $$G_{I}$$ for the matrix formed by the columns of $$G$$ indexed by $$I$$. The Lee-Brickell ISD loops the following until it is successful:

1. Choose an information set $$I$$ of $$C$$.

2. Compute $$r' = r - r_{I}\times G_I^{-1} \times G$$

3. Consider every size-$$p$$ subset of $$I$$, $$\{a_1, \dots, a_p\}$$. For each $$m = (m_1, \dots, m_p) \in GF(q)^{p}$$, compute the error vector $$e = r' - \sum_{i=1}^{p} m_i\times g_{a_i}$$,

4. If $$e$$ has a Hamming weight at most $$w$$, return $$r-e$$.

INPUT:

• code – A linear code for which to decode.

• decoding_interval – a pair of integers specifying an interval of number of errors to correct. Includes both end values.

• search_size – (optional) the size of subsets to use on step 3 of the algorithm as described above. Usually a small number. It has to be at most the largest allowed number of errors. A good choice will be approximated if this option is not set; see sage.coding.LeeBrickellISDAlgorithm.calibrate() for details.

EXAMPLES:

sage: C = codes.GolayCode(GF(2))
sage: from sage.coding.information_set_decoder import LeeBrickellISDAlgorithm
sage: A = LeeBrickellISDAlgorithm(C, (0,4)); A
ISD Algorithm (Lee-Brickell) for [24, 12, 8] Extended Golay code over GF(2) decoding up to 4 errors

sage: C = codes.GolayCode(GF(2))
sage: A = LeeBrickellISDAlgorithm(C, (2,3)); A
ISD Algorithm (Lee-Brickell) for [24, 12, 8] Extended Golay code over GF(2) decoding between 2 and 3 errors

calibrate()

Run some test computations to estimate the optimal search size.

Let $$p$$ be the search size. We should simply choose $$p$$ such that the average expected time is minimal. The algorithm succeeds when it chooses an information set with at least $$k - p$$ correct positions, where $$k$$ is the dimension of the code and $$p$$ the search size. The expected number of trials we need before this occurs is:

$\binom{n}{k}/(\rho \sum_{i=0}^p \binom{n-\tau}{k-i} \binom{\tau}{i})$

Here $$\rho$$ is the fraction of $$k$$ subsets of indices which are information sets. If $$T$$ is the average time for steps 1 and 2 (including selecting $$I$$ until an information set is found), while $$P(i)$$ is the time for the body of the for-loop in step 3 for $$m$$ of weight $$i$$, then each information set trial takes roughly time $$T + \sum_{i=0}^{p} P(i) \binom{k}{i} (q-1)^i$$, where $$\GF{q}$$ is the base field.

The values $$T$$ and $$P$$ are here estimated by running a few test computations similar to those done by the decoding algorithm. We don’t explicitly estimate $$\rho$$.

OUTPUT: Does not output anything but sets private fields used by sage.coding.information_set_decoder.InformationSetAlgorithm.parameters() and sage.coding.information_set_decoder.InformationSetAlgorithm.time_estimate()().

EXAMPLES:

sage: from sage.coding.information_set_decoder import LeeBrickellISDAlgorithm
sage: C = codes.GolayCode(GF(2))
sage: A = LeeBrickellISDAlgorithm(C, (0,3)); A
ISD Algorithm (Lee-Brickell) for [24, 12, 8] Extended Golay code over GF(2) decoding up to 3 errors
sage: A.calibrate()
sage: A.parameters() #random
{'search_size': 1}
sage: A.time_estimate() #random
0.0008162108571427874


If we specify the parameter at construction time, calibrate does not override this choice:

sage: A = LeeBrickellISDAlgorithm(C, (0,3), search_size=2); A
ISD Algorithm (Lee-Brickell) for [24, 12, 8] Extended Golay code over GF(2) decoding up to 3 errors
sage: A.parameters()
{'search_size': 2}
sage: A.calibrate()
sage: A.parameters()
{'search_size': 2}
sage: A.time_estimate() #random
0.0008162108571427874

decode(r)

The Lee-Brickell algorithm as described in the class doc.

Note that either parameters must be given at construction time or sage.coding.information_set_decoder.InformationSetAlgorithm.calibrate() should be called before calling this method.

INPUT:

• $$r$$ – a received word, i.e. a vector in the ambient space of decoder.Decoder.code().

OUTPUT: A codeword whose distance to $$r$$ satisfies self.decoding_interval().

EXAMPLES:

sage: M = matrix(GF(2), [[1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0],\
[0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1],\
[0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0],\
[0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1],\
[0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1]])
sage: C = codes.LinearCode(M)
sage: from sage.coding.information_set_decoder import LeeBrickellISDAlgorithm
sage: A = LeeBrickellISDAlgorithm(C, (2,2))
sage: c = C.random_element()
sage: Chan = channels.StaticErrorRateChannel(C.ambient_space(), 2)
sage: r = Chan(c)
sage: c_out = A.decode(r)
sage: (r - c).hamming_weight() == 2
True

class sage.coding.information_set_decoder.LinearCodeInformationSetDecoder(code, number_errors, algorithm=None, **kwargs)

Information-set decoder for any linear code.

Information-set decoding is a probabilistic decoding strategy that essentially tries to guess $$k$$ correct positions in the received word, where $$k$$ is the dimension of the code. A codeword agreeing with the received word on the guessed position can easily be computed, and their difference is one possible error vector. A “correct” guess is assumed when this error vector has low Hamming weight.

The ISD strategy requires choosing how many errors is deemed acceptable. One choice could be $$d/2$$, where $$d$$ is the minimum distance of the code, but sometimes $$d$$ is not known, or sometimes more errors are expected. If one chooses anything above $$d/2$$, the algorithm does not guarantee to return a nearest codeword.

This simple algorithm is not very efficient in itself, but there are numerous refinements to the strategy. Specifying which strategy to use among those that Sage knows is done using the algorithm keyword. If this is not set, an efficient choice will be made for you.

The various ISD algorithms all need to select a number of parameters. If you choose a specific algorithm to use, you can pass these parameters as named parameters directly to this class’ constructor. If you don’t, efficient choices will be calibrated for you.

Warning

If there is no codeword within the specified decoding distance, then the decoder may never terminate, or it may raise a sage.coding.decoder.DecodingError exception, depending on the ISD algorithm used.

INPUT:

• code – A linear code for which to decode.

• number_errors – an integer, the maximal number of errors to accept as correct decoding. An interval can also be specified by giving a pair of integers, where both end values are taken to be in the interval.

• algorithm – (optional) the string name of the ISD algorithm to employ. If this is not set, an appropriate one will be chosen. A constructed sage.coding.information_set_decoder.InformationSetAlgorithm object may also be given. In this case number_errors must match that of the passed algorithm.

• **kwargs – (optional) any number of named arguments passed on to the ISD algorithm. Such are usually not required, and they can only be set if algorithm is set to a specific algorithm. See the documentation for each individual ISD algorithm class for information on any named arguments they may accept. The easiest way to access this documentation is to first construct the decoder without passing any named arguments, then accessing the ISD algorithm using sage.coding.information_set_decoder.LinearCodeInformationSetDecoder.algorithm(), and then reading the $$?$$ help on the constructed object.

EXAMPLES:

The principal way to access this class is through the sage.code.linear_code.AbstractLinearCode.decoder() method:

sage: C = codes.GolayCode(GF(3))
sage: D = C.decoder("InformationSet", 2); D
Information-set decoder (Lee-Brickell) for [12, 6, 6] Extended Golay code over GF(3) decoding up to 2 errors


You can specify which algorithm you wish to use, and you should do so in order to pass special parameters to it:

sage: C = codes.GolayCode(GF(3))
sage: D2 = C.decoder("InformationSet", 2, algorithm="Lee-Brickell", search_size=2); D2
Information-set decoder (Lee-Brickell) for [12, 6, 6] Extended Golay code over GF(3) decoding up to 2 errors
sage: D2.algorithm()
ISD Algorithm (Lee-Brickell) for [12, 6, 6] Extended Golay code over GF(3) decoding up to 2 errors
sage: D2.algorithm().parameters()
{'search_size': 2}


If you specify an algorithm which is not known, you get a friendly error message:

sage: C.decoder("InformationSet", 2, algorithm="NoSuchThing")
Traceback (most recent call last):
...
ValueError: Unknown ISD algorithm 'NoSuchThing'. The known algorithms are ['Lee-Brickell'].


You can also construct an ISD algorithm separately and pass that. This is mostly useful if you write your own ISD algorithms:

sage: from sage.coding.information_set_decoder import LeeBrickellISDAlgorithm
sage: A = LeeBrickellISDAlgorithm(C, (0, 2))
sage: D = C.decoder("InformationSet", 2, algorithm=A); D
Information-set decoder (Lee-Brickell) for [12, 6, 6] Extended Golay code over GF(3) decoding up to 2 errors


When passing an already constructed ISD algorithm, you can’t also pass parameters to the ISD algorithm when constructing the decoder:

sage: C.decoder("InformationSet", 2, algorithm=A, search_size=2)
Traceback (most recent call last):
...
ValueError: ISD algorithm arguments are not allowed when supplying a constructed ISD algorithm


We can also information-set decode non-binary codes:

sage: C = codes.GolayCode(GF(3))
sage: D = C.decoder("InformationSet", 2); D
Information-set decoder (Lee-Brickell) for [12, 6, 6] Extended Golay code over GF(3) decoding up to 2 errors


There are two other ways to access this class:

sage: D = codes.decoders.LinearCodeInformationSetDecoder(C, 2); D
Information-set decoder (Lee-Brickell) for [12, 6, 6] Extended Golay code over GF(3) decoding up to 2 errors

sage: from sage.coding.information_set_decoder import LinearCodeInformationSetDecoder
sage: D = LinearCodeInformationSetDecoder(C, 2); D
Information-set decoder (Lee-Brickell) for [12, 6, 6] Extended Golay code over GF(3) decoding up to 2 errors

algorithm()

Return the ISD algorithm used by this ISD decoder.

EXAMPLES:

sage: C = codes.GolayCode(GF(2))
sage: D = C.decoder("InformationSet", (2,4), "Lee-Brickell")
sage: D.algorithm()
ISD Algorithm (Lee-Brickell) for [24, 12, 8] Extended Golay code over GF(2) decoding between 2 and 4 errors

decode_to_code(r)

Decodes a received word with respect to the associated code of this decoder.

Warning

If there is no codeword within the decoding radius of this decoder, this method may never terminate, or it may raise a sage.coding.decoder.DecodingError exception, depending on the ISD algorithm used.

INPUT:

• r – a vector in the ambient space of decoder.Decoder.code().

OUTPUT: a codeword of decoder.Decoder.code().

EXAMPLES:

sage: M = matrix(GF(2), [[1,0,0,0,0,0,1,0,1,0,1,1,0,0,1],\
[0,1,0,0,0,1,1,1,1,0,0,0,0,1,1],\
[0,0,1,0,0,0,0,1,0,1,1,1,1,1,0],\
[0,0,0,1,0,0,1,0,1,0,0,0,1,1,0],\
[0,0,0,0,1,0,0,0,1,0,1,1,0,1,0]])
sage: C = LinearCode(M)
sage: c = C.random_element()
sage: Chan = channels.StaticErrorRateChannel(C.ambient_space(), 2)
sage: r = Chan(c)
sage: D = C.decoder('InformationSet', 2)
sage: c == D.decode_to_code(r)
True


Information-set decoding a non-binary code:

sage: C = codes.GolayCode(GF(3)); C
[12, 6, 6] Extended Golay code over GF(3)
sage: c = C.random_element()
sage: Chan = channels.StaticErrorRateChannel(C.ambient_space(), 2)
sage: r = Chan(c)
sage: D = C.decoder('InformationSet', 2)
sage: c == D.decode_to_code(r)
True


Let’s take a bigger example, for which syndrome decoding or nearest-neighbor decoding would be infeasible: the $$[59, 30]$$ Quadratic Residue code over $$\GF{3}$$ has true minimum distance 17, so we can correct 8 errors:

sage: C = codes.QuadraticResidueCode(59, GF(3))
sage: c = C.random_element()
sage: Chan = channels.StaticErrorRateChannel(C.ambient_space(), 2)
sage: r = Chan(c)
sage: D = C.decoder('InformationSet', 8)
sage: c == D.decode_to_code(r) # long time
True

decoding_interval()

A pair of integers specifying the interval of number of errors this decoder will attempt to correct.

The interval includes both end values.

EXAMPLES:

sage: C = codes.GolayCode(GF(2))
sage: D = C.decoder("InformationSet", 2)
sage: D.decoding_interval()
(0, 2)

decoding_radius()

Return the maximal number of errors this decoder can decode.

EXAMPLES:

sage: C = codes.GolayCode(GF(2))
sage: D = C.decoder("InformationSet", 2)
sage: D.decoding_radius()
2

static known_algorithms(dictionary=False)

Return the list of ISD algorithms that Sage knows.

Passing any of these to the constructor of sage.coding.information_set_decoder.LinearCodeInformationSetDecoder will make the ISD decoder use that algorithm.

INPUT:

• dictionary - optional. If set to True, return a dict mapping decoding algorithm name to its class.

OUTPUT: a list of strings or a dict from string to ISD algorithm class.

EXAMPLES:

sage: from sage.coding.information_set_decoder import LinearCodeInformationSetDecoder
sage: sorted(LinearCodeInformationSetDecoder.known_algorithms())
['Lee-Brickell']
`