Generic structures for linear codes over the rank metric¶
Rank Metric¶
In coding theory, the most common metric is the Hamming metric, where distance between two codewords is given by the number of positions in which they differ. An alternative to this is the rank metric. Take two fields, \(\GF{q}\) and \(\GF{q^m}\), and define a code \(C\) to be a set of vectors of length \(n\) with entries from \(\GF{q^m}\). Let \(c\) be a codeword. We can represent it as an \(m \times n\) matrix \(M\) over \(\GF{q}\).
A detailed description on the relationship between the two representations can
be found in sage.coding.linear_rank_metric.to_matrix_representation()
and sage.coding.linear_rank_metric.from_matrix_representation().
We can define a metric using the rank of the matrix representation of the codewords. A distance between two codewords \(a, b\) is the rank of the matrix representation of \(a - b\). A weight of a codeword \(c\) is the rank of the matrix representation of \(c\).
This module allows representing rank metric codes which are linear over the big field \(\GF{q^m}\), i.e. the usual linearity condition when the codewords are considered in vector form. One can also consider rank metric codes which are only linear over \(\GF{q}\), but these are not currently supported in SageMath.
Note that linear rank metric codes per the definition of this file are
mathematically just linear block codes, and so could be considered as a
sage.coding.linear_code.LinearCode. However, since most of the
functionality of that class is specific to the Hamming metric, the two notions
are implemented as entirely different in SageMath. If you wish to investigate
Hamming-metric properties of a linear rank metric code C, you can easily
convert it by calling C_hamm = LinearCode(C).
Linear Rank Metric Code and Gabidulin Codes¶
The class sage.coding.linear_rank_metric.LinearRankMetricCode is the
analog of sage.coding.linear_code.LinearCode, i.e. it is a generator
matrix-based representation of a linear rank metric code without specific
knowledge on the structure of the code.
Gabidulin codes are the main family of structured linear rank metric codes. These codes are the rank-metric analog of Reed-Solomon codes.
AbstractLinearRankMetricCode¶
This is a base class designed to contain methods, features and parameters shared by every linear rank metric code. For instance, generic algorithms for computing the minimum distance, etc. Many of these algorithms are slow, e.g. exponential in the code length. It also contains methods for swapping between vector and matrix representation of elements.
AbstractLinearCodeNoMetric is an abstract class for linear rank metric codes,
so any linear rank metric code  class should inherit from this class.
Also AbstractLinearCodeNoMetric should never itself be instantiated.
See sage.coding.linear_rank_metric.AbstractLinearRankMetricCode
for details and examples.
LinearRankMetricCode¶
This class is used to represent arbitrary and unstructured linear rank metric
codes. It mostly relies directly on generic methods provided by
AbstractLinearRankMetricCode, which means that basic operations on the code
(e.g. computation of the minimum distance) will use slow algorithms.
A LinearRankMetricCode is instantiated by providing a generator:
sage: G = Matrix(GF(64), [[1,1,0], [0,0,1]])
sage: C = codes.LinearRankMetricCode(G, GF(4))
sage: C
[3, 2] linear rank metric code over GF(64)/GF(4)
sage: C.generator_matrix()
[1 1 0]
[0 0 1]
sage: c = vector(GF(64), (1, 1, 1))
sage: c in C
True
>>> from sage.all import *
>>> G = Matrix(GF(Integer(64)), [[Integer(1),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(1)]])
>>> C = codes.LinearRankMetricCode(G, GF(Integer(4)))
>>> C
[3, 2] linear rank metric code over GF(64)/GF(4)
>>> C.generator_matrix()
[1 1 0]
[0 0 1]
>>> c = vector(GF(Integer(64)), (Integer(1), Integer(1), Integer(1)))
>>> c in C
True
Further references¶
Read more about rank metric and Gabidulin codes
AUTHORS:
- Marketa Slukova (2019-08-16): initial version 
- Camille Garnier and Rubén Muñoz--Bertrand (2024-02-13): added rank_support_of_vector, and corrected the documentation 
- class sage.coding.linear_rank_metric.AbstractLinearRankMetricCode(base_field, sub_field, length, default_encoder_name, default_decoder_name, basis=None)[source]¶
- Bases: - AbstractLinearCodeNoMetric- Abstract class for linear rank metric codes. - This class contains methods that can be used on families of linear rank metric codes. Every linear rank metric code class should inherit from this abstract class. - This class is intended for codes which are linear over the - base_field.- Codewords of rank metric codes have two representations. They can either be written as a vector of length \(n\) over \(\GF{q^m}\), or an \(m \times n\) matrix over \(\GF{q}\). This implementation principally uses the vector representation. However, one can always get the matrix representation using the - sage.coding.linear_rank_metric.AbstractLinearRankMetricCode.to_matrix()method. To go back to a vector, use the- sage.coding.linear_rank_metric.AbstractLinearRankMetricCode.from_matrix()method.- Instructions on how to make a new family of rank metric codes is analogous to making a new family of linear codes over the Hamming metric, instructions for which are in - sage.coding.linear_code.AbstractLinearCode. For an example on, see- sage.coding.linear_rank_metric.AbstractLinearRankMetricCode.__init__()- Warning - A lot of methods of the abstract class rely on the knowledge of a generator matrix. It is thus strongly recommended to set an encoder with a generator matrix implemented as a default encoder. - extension_degree()[source]¶
- Return \(m\), the degree of the field extension of - self.- Let - base_fieldbe \(\GF{q^m}\) and- sub_fieldbe \(\GF{q}\). Then this function returns \(m\).- EXAMPLES: - sage: G = Matrix(GF(64), [[1,1,0], [0,0,1]]) sage: C = codes.LinearRankMetricCode(G, GF(4)) sage: C.extension_degree() 3 - >>> from sage.all import * >>> G = Matrix(GF(Integer(64)), [[Integer(1),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(1)]]) >>> C = codes.LinearRankMetricCode(G, GF(Integer(4))) >>> C.extension_degree() 3 
 - field_extension()[source]¶
- Return the field extension of - self.- Let - base_fieldbe some field \(\GF{q^m}\) and- sub_field\(\GF{q}\). This function returns the vector space of dimension \(m\) over \(\GF{q}\).- EXAMPLES: - sage: G = Matrix(GF(64), [[1,1,0], [0,0,1]]) sage: C = codes.LinearRankMetricCode(G, GF(4)) sage: C.field_extension() Vector space of dimension 3 over Finite Field in z2 of size 2^2 - >>> from sage.all import * >>> G = Matrix(GF(Integer(64)), [[Integer(1),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(1)]]) >>> C = codes.LinearRankMetricCode(G, GF(Integer(4))) >>> C.field_extension() Vector space of dimension 3 over Finite Field in z2 of size 2^2 
 - matrix_form_of_vector(word)[source]¶
- Return the matrix representation of a word. - INPUT: - word– a vector over the- base_fieldof- self
 - EXAMPLES: - sage: G = Matrix(GF(64), [[1,1,0], [0,0,1]]) sage: C = codes.LinearRankMetricCode(G, GF(4)) sage: x = GF(64).gen() sage: a = vector(GF(64), (x + 1, x + 1, 1)) sage: C.matrix_form_of_vector(a) [1 1 1] [1 1 0] [0 0 0] - >>> from sage.all import * >>> G = Matrix(GF(Integer(64)), [[Integer(1),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(1)]]) >>> C = codes.LinearRankMetricCode(G, GF(Integer(4))) >>> x = GF(Integer(64)).gen() >>> a = vector(GF(Integer(64)), (x + Integer(1), x + Integer(1), Integer(1))) >>> C.matrix_form_of_vector(a) [1 1 1] [1 1 0] [0 0 0] 
 - minimum_distance()[source]¶
- Return the minimum distance of - self.- This algorithm simply iterates over all the elements of the code and returns the minimum weight. - EXAMPLES: - sage: F.<a> = GF(8) sage: G = Matrix(F, [[1,a,a^2,0]]) sage: C = codes.LinearRankMetricCode(G, GF(2)) sage: C.minimum_distance() 3 - >>> from sage.all import * >>> F = GF(Integer(8), names=('a',)); (a,) = F._first_ngens(1) >>> G = Matrix(F, [[Integer(1),a,a**Integer(2),Integer(0)]]) >>> C = codes.LinearRankMetricCode(G, GF(Integer(2))) >>> C.minimum_distance() 3 
 - rank_distance_between_vectors(left, right)[source]¶
- Return the rank of the matrix of - left-- right.- INPUT: - left– a vector over the- base_fieldof- self
- right– a vector over the- base_fieldof- self
 - EXAMPLES: - sage: G = Matrix(GF(64), [[1,1,0], [0,0,1]]) sage: C = codes.LinearRankMetricCode(G, GF(4)) sage: x = GF(64).gen() sage: a = vector(GF(64), (x + 1, x + 1, 1)) sage: b = vector(GF(64), (1, 0, 0)) sage: C.rank_distance_between_vectors(a, b) 2 - >>> from sage.all import * >>> G = Matrix(GF(Integer(64)), [[Integer(1),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(1)]]) >>> C = codes.LinearRankMetricCode(G, GF(Integer(4))) >>> x = GF(Integer(64)).gen() >>> a = vector(GF(Integer(64)), (x + Integer(1), x + Integer(1), Integer(1))) >>> b = vector(GF(Integer(64)), (Integer(1), Integer(0), Integer(0))) >>> C.rank_distance_between_vectors(a, b) 2 
 - rank_support_of_vector(word, sub_field=None, basis=None)[source]¶
- Return the rank support of - wordover- sub_field, i.e. the vector space over- sub_fieldgenerated by its coefficients.- If - wordis a vector over some field \(\GF{q^m}\), and- sub_fieldis a subfield of \(\GF{q^m}\), the function converts it into a matrix over- sub_field, with respect to the basis- basis.- INPUT: - word– a vector over the- base_fieldof- self.
- sub_field– (default:- None) a sub field of the- base_fieldof- self; if not specified, it is the prime subfield of \(\GF{p}\) the- base_fieldof- self.
- basis– (default:- None) a basis of- base_fieldof- selfas a vector space over- sub_field. If not specified, the default basis is \(1,\beta,\ldots,\beta^{m-1}\), where \(\beta\) is the generator of \(\GF{q^m}\) given by SageMath.
 - EXAMPLES: - sage: G = Matrix(GF(64), [[1,1,0], [0,0,1]]) sage: C = codes.LinearRankMetricCode(G, GF(4)) sage: a = GF(64).gen() sage: c = vector([a^4 + a^3 + 1, a^4 + a^3 + 1, a^4 + a^3 + a^2 + 1]) sage: c in C True sage: C.rank_support_of_vector(c) Vector space of degree 6 and dimension 2 over Finite Field of size 2 Basis matrix: [1 0 0 1 1 0] [0 0 1 0 0 0] - >>> from sage.all import * >>> G = Matrix(GF(Integer(64)), [[Integer(1),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(1)]]) >>> C = codes.LinearRankMetricCode(G, GF(Integer(4))) >>> a = GF(Integer(64)).gen() >>> c = vector([a**Integer(4) + a**Integer(3) + Integer(1), a**Integer(4) + a**Integer(3) + Integer(1), a**Integer(4) + a**Integer(3) + a**Integer(2) + Integer(1)]) >>> c in C True >>> C.rank_support_of_vector(c) Vector space of degree 6 and dimension 2 over Finite Field of size 2 Basis matrix: [1 0 0 1 1 0] [0 0 1 0 0 0] - An example with a non canonical basis: - sage: K.<a> = GF(2^3) sage: G = Matrix(K, [[1,1,0], [0,0,1]]) sage: C = codes.LinearRankMetricCode(G) sage: c = vector([a^2, a^2, 0]) sage: basis = [a, a+1, a^2] sage: C.rank_support_of_vector(c, basis=basis) Vector space of degree 3 and dimension 1 over Finite Field of size 2 Basis matrix: [0 0 1] - >>> from sage.all import * >>> K = GF(Integer(2)**Integer(3), names=('a',)); (a,) = K._first_ngens(1) >>> G = Matrix(K, [[Integer(1),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(1)]]) >>> C = codes.LinearRankMetricCode(G) >>> c = vector([a**Integer(2), a**Integer(2), Integer(0)]) >>> basis = [a, a+Integer(1), a**Integer(2)] >>> C.rank_support_of_vector(c, basis=basis) Vector space of degree 3 and dimension 1 over Finite Field of size 2 Basis matrix: [0 0 1] 
 - rank_weight_of_vector(word)[source]¶
- Return the weight of the word, i.e. its rank. - INPUT: - word– a vector over the- base_fieldof- self
 - EXAMPLES: - sage: G = Matrix(GF(64), [[1,1,0], [0,0,1]]) sage: C = codes.LinearRankMetricCode(G, GF(4)) sage: x = GF(64).gen() sage: a = vector(GF(64), (x + 1, x + 1, 1)) sage: C.rank_weight_of_vector(a) 2 - >>> from sage.all import * >>> G = Matrix(GF(Integer(64)), [[Integer(1),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(1)]]) >>> C = codes.LinearRankMetricCode(G, GF(Integer(4))) >>> x = GF(Integer(64)).gen() >>> a = vector(GF(Integer(64)), (x + Integer(1), x + Integer(1), Integer(1))) >>> C.rank_weight_of_vector(a) 2 
 - sub_field()[source]¶
- Return the sub field of - self.- EXAMPLES: - sage: G = Matrix(GF(64), [[1,1,0], [0,0,1]]) sage: C = codes.LinearRankMetricCode(G, GF(4)) sage: C.sub_field() Finite Field in z2 of size 2^2 - >>> from sage.all import * >>> G = Matrix(GF(Integer(64)), [[Integer(1),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(1)]]) >>> C = codes.LinearRankMetricCode(G, GF(Integer(4))) >>> C.sub_field() Finite Field in z2 of size 2^2 
 - vector_form_of_matrix(word)[source]¶
- Return the vector representation of a word. - INPUT: - word– a matrix over the- sub_fieldof- self
 - EXAMPLES: - sage: G = Matrix(GF(64), [[1,1,0], [0,0,1]]) sage: C = codes.LinearRankMetricCode(G, GF(4)) sage: x = GF(64).gen() sage: m = Matrix(GF(4), [[1, 1, 1], [1, 1, 0], [0, 0, 0]]) sage: C.vector_form_of_matrix(m) (z6 + 1, z6 + 1, 1) - >>> from sage.all import * >>> G = Matrix(GF(Integer(64)), [[Integer(1),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(1)]]) >>> C = codes.LinearRankMetricCode(G, GF(Integer(4))) >>> x = GF(Integer(64)).gen() >>> m = Matrix(GF(Integer(4)), [[Integer(1), Integer(1), Integer(1)], [Integer(1), Integer(1), Integer(0)], [Integer(0), Integer(0), Integer(0)]]) >>> C.vector_form_of_matrix(m) (z6 + 1, z6 + 1, 1) 
 
- class sage.coding.linear_rank_metric.LinearRankMetricCode(generator, sub_field=None, basis=None)[source]¶
- Bases: - AbstractLinearRankMetricCode- Linear rank metric codes over a finite field, represented using a generator matrix. - This class should be used for arbitrary and unstructured linear rank metric codes. This means that basic operations on the code, such as the computation of the minimum distance, will use generic, slow algorithms. - If you are looking for constructing a code from a more specific family, see if the family has been implemented by investigating - codes.<tab>. These more specific classes use properties particular to that family to allow faster algorithms, and could also have family-specific methods.- EXAMPLES: - sage: G = Matrix(GF(64), [[1,1,0], [0,0,1]]) sage: C = codes.LinearRankMetricCode(G, GF(4)) sage: C [3, 2] linear rank metric code over GF(64)/GF(4) sage: C.base_field() Finite Field in z6 of size 2^6 sage: C.sub_field() Finite Field in z2 of size 2^2 sage: C.length() 3 sage: C.dimension() 2 sage: C[2] (z6, z6, 0) sage: E = codes.encoders.LinearCodeGeneratorMatrixEncoder(C) sage: word = vector(C.base_field(), [1, 0]) sage: E(word) (1, 1, 0) - >>> from sage.all import * >>> G = Matrix(GF(Integer(64)), [[Integer(1),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(1)]]) >>> C = codes.LinearRankMetricCode(G, GF(Integer(4))) >>> C [3, 2] linear rank metric code over GF(64)/GF(4) >>> C.base_field() Finite Field in z6 of size 2^6 >>> C.sub_field() Finite Field in z2 of size 2^2 >>> C.length() 3 >>> C.dimension() 2 >>> C[Integer(2)] (z6, z6, 0) >>> E = codes.encoders.LinearCodeGeneratorMatrixEncoder(C) >>> word = vector(C.base_field(), [Integer(1), Integer(0)]) >>> E(word) (1, 1, 0) - generator_matrix(encoder_name=None, **kwargs)[source]¶
- Return a generator matrix of - self.- INPUT: - encoder_name– (default:- None) name of the encoder which will be used to compute the generator matrix.- self._generator_matrixwill be returned if default value is kept.
- kwargs– all additional arguments are forwarded to the construction of the encoder that is used
 - EXAMPLES: - sage: G = Matrix(GF(64), [[1,1,0], [0,0,1]]) sage: C = codes.LinearRankMetricCode(G, GF(4)) sage: C.generator_matrix() [1 1 0] [0 0 1] - >>> from sage.all import * >>> G = Matrix(GF(Integer(64)), [[Integer(1),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(1)]]) >>> C = codes.LinearRankMetricCode(G, GF(Integer(4))) >>> C.generator_matrix() [1 1 0] [0 0 1] 
 
- class sage.coding.linear_rank_metric.LinearRankMetricCodeNearestNeighborDecoder(code)[source]¶
- Bases: - Decoder- Construct a decoder for Linear Rank Metric Codes. - This decoder will decode to the nearest codeword found. - decode_to_code(r)[source]¶
- Corrects the errors in - wordand returns a codeword.- INPUT: - r– a codeword of- self
 - OUTPUT: a vector of - self’s message space- EXAMPLES: - sage: F.<a> = GF(4) sage: G = Matrix(F, [[1,1,0]]) sage: C = codes.LinearRankMetricCode(G, GF(2)) sage: D = codes.decoders.LinearRankMetricCodeNearestNeighborDecoder(C) sage: D.decode_to_code(vector(F, [a, a, 1])) (a, a, 0) - >>> from sage.all import * >>> F = GF(Integer(4), names=('a',)); (a,) = F._first_ngens(1) >>> G = Matrix(F, [[Integer(1),Integer(1),Integer(0)]]) >>> C = codes.LinearRankMetricCode(G, GF(Integer(2))) >>> D = codes.decoders.LinearRankMetricCodeNearestNeighborDecoder(C) >>> D.decode_to_code(vector(F, [a, a, Integer(1)])) (a, a, 0) 
 - decoding_radius()[source]¶
- Return maximal number of errors - selfcan decode.- EXAMPLES: - sage: F.<a> = GF(8) sage: G = Matrix(F, [[1,a,a^2,0]]) sage: C = codes.LinearRankMetricCode(G, GF(2)) sage: D = codes.decoders.LinearRankMetricCodeNearestNeighborDecoder(C) sage: D.decoding_radius() 1 - >>> from sage.all import * >>> F = GF(Integer(8), names=('a',)); (a,) = F._first_ngens(1) >>> G = Matrix(F, [[Integer(1),a,a**Integer(2),Integer(0)]]) >>> C = codes.LinearRankMetricCode(G, GF(Integer(2))) >>> D = codes.decoders.LinearRankMetricCodeNearestNeighborDecoder(C) >>> D.decoding_radius() 1 
 
- sage.coding.linear_rank_metric.from_matrix_representation(w, base_field=None, basis=None)[source]¶
- Return a vector representation of a matrix - wover- base_fieldin terms of- basis.- Given an \(m \times n\) matrix over \(\GF{q}\) and some - basisof \(\GF{q^m}\) over \(\GF{q}\), we can represent each of its columns as an element of \(\GF{q^m}\), yielding a vector of length \(n\) over \(\GF{q}\).- In case - base_fieldis not given, we take \(\GF{q^m}\), the field extension of \(\GF{q}\) of degree \(m\), the number of rows of- w.- INPUT: - w– a matrix over some field \(\GF{q}\)
- base_field– (default:- None) an extension field of \(\GF{q}\). If not specified, it is the field \(\GF{q^m}\), where \(m\) is the number of rows of- w.
- basis– (default:- None) a basis of \(\GF{q^m}\) as a vector space over \(\GF{q}\). If not specified, the default basis is \(1,\beta,\ldots,\beta^{m-1}\) where \(\beta\) is the generator of \(\GF{q^m}\) given by SageMath.
 - EXAMPLES: - sage: from sage.coding.linear_rank_metric import from_matrix_representation sage: m = Matrix(GF(4), [[1, 1, 1], [1, 1, 0], [0, 0, 0]]) sage: from_matrix_representation(m) (z6 + 1, z6 + 1, 1) sage: v = vector(GF(4), (1, 0, 0)) sage: from_matrix_representation(v) Traceback (most recent call last): ... TypeError: Input must be a matrix - >>> from sage.all import * >>> from sage.coding.linear_rank_metric import from_matrix_representation >>> m = Matrix(GF(Integer(4)), [[Integer(1), Integer(1), Integer(1)], [Integer(1), Integer(1), Integer(0)], [Integer(0), Integer(0), Integer(0)]]) >>> from_matrix_representation(m) (z6 + 1, z6 + 1, 1) >>> v = vector(GF(Integer(4)), (Integer(1), Integer(0), Integer(0))) >>> from_matrix_representation(v) Traceback (most recent call last): ... TypeError: Input must be a matrix 
- sage.coding.linear_rank_metric.rank_distance(a, b, sub_field=None, basis=None)[source]¶
- Return the rank of - a-- bas a matrix over- sub_field.- Take two vectors - a,- bover some field \(\GF{q^m}\). This function converts them to matrices over \(\GF{q}\) and calculates the rank of their difference.- If - sub_fieldis not specified, we take the prime subfield \(\GF{q}\) of \(\GF{q^m}\).- INPUT: - a– a vector over some field \(\GF{q^m}\)
- b– a vector over some field \(\GF{q^m}\)
- sub_field– (default:- None) a sub field of \(\GF{q^m}\); if not specified, it is the prime subfield \(\GF{p}\) of \(\GF{q^m}\)
- basis– (default:- None) a basis of \(\GF{q^m}\) as a vector space over- sub_field. If not specified, the default basis is \(1,\beta,\ldots,\beta^{m-1}\) where \(\beta\) is the generator of \(\GF{q^m}\) given by SageMath.
 - EXAMPLES: - sage: from sage.coding.linear_rank_metric import rank_distance sage: x = GF(64).gen() sage: a = vector(GF(64), (x + 1, x + 1, 1)) sage: b = vector(GF(64), (1, 0, 0)) sage: rank_distance(a, b, GF(4)) 2 sage: c = vector(GF(4), (1, 0, 0)) sage: rank_distance(a, c, GF(4)) Traceback (most recent call last): ... ValueError: The base field of (z6 + 1, z6 + 1, 1) and (1, 0, 0) has to be the same sage: d = Matrix(GF(64), (1, 0, 0)) sage: rank_distance(a, d, GF(64)) Traceback (most recent call last): ... TypeError: Both inputs have to be vectors sage: e = vector(GF(64), (1, 0)) sage: rank_distance(a, e, GF(64)) Traceback (most recent call last): ... ValueError: The length of (z6 + 1, z6 + 1, 1) and (1, 0) has to be the same - >>> from sage.all import * >>> from sage.coding.linear_rank_metric import rank_distance >>> x = GF(Integer(64)).gen() >>> a = vector(GF(Integer(64)), (x + Integer(1), x + Integer(1), Integer(1))) >>> b = vector(GF(Integer(64)), (Integer(1), Integer(0), Integer(0))) >>> rank_distance(a, b, GF(Integer(4))) 2 >>> c = vector(GF(Integer(4)), (Integer(1), Integer(0), Integer(0))) >>> rank_distance(a, c, GF(Integer(4))) Traceback (most recent call last): ... ValueError: The base field of (z6 + 1, z6 + 1, 1) and (1, 0, 0) has to be the same >>> d = Matrix(GF(Integer(64)), (Integer(1), Integer(0), Integer(0))) >>> rank_distance(a, d, GF(Integer(64))) Traceback (most recent call last): ... TypeError: Both inputs have to be vectors >>> e = vector(GF(Integer(64)), (Integer(1), Integer(0))) >>> rank_distance(a, e, GF(Integer(64))) Traceback (most recent call last): ... ValueError: The length of (z6 + 1, z6 + 1, 1) and (1, 0) has to be the same 
- sage.coding.linear_rank_metric.rank_weight(c, sub_field=None, basis=None)[source]¶
- Return the rank of - cas a matrix over- sub_field.- If - cis a vector over some field \(\GF{q^m}\), the function converts it into a matrix over- sub_field`.- INPUT: - c– a vector over some field \(\GF{q^m}\); or a matrix over \(\GF{q}\)
- sub_field– (default:- None) a sub field of \(\GF{q^m}\); if not specified, it is the prime subfield \(\GF{p}\) of \(\GF{q^m}\)
- basis– (default:- None) a basis of \(\GF{q^m}\) as a vector space over- sub_field. If not specified, the default basis is \(1,\beta,\ldots,\beta^{m-1}\) where \(\beta\) is the generator of \(\GF{q^m}\) given by SageMath.
 - EXAMPLES: - sage: from sage.coding.linear_rank_metric import rank_weight sage: x = GF(64).gen() sage: a = vector(GF(64), (x + 1, x + 1, 1)) sage: rank_weight(a, GF(4)) 2 - >>> from sage.all import * >>> from sage.coding.linear_rank_metric import rank_weight >>> x = GF(Integer(64)).gen() >>> a = vector(GF(Integer(64)), (x + Integer(1), x + Integer(1), Integer(1))) >>> rank_weight(a, GF(Integer(4))) 2 
- sage.coding.linear_rank_metric.to_matrix_representation(v, sub_field=None, basis=None)[source]¶
- Return a matrix representation of - vover- sub_fieldin terms of- basis.- Let \((b_1, b_2, \ldots, b_m)\), \(b_i \in GF(q^m)\), be a basis of \(\GF{q^m}\) as a vector space over \(\GF{q}\). Take an element \(x \in \GF{q^m}\). We can write \(x\) as \(x = u_1 b_1 + u_2 b_2 + \ldots u_m b_m\), where \(u_i \in GF(q)\). This way we can represent an element from \(\GF{q^m}\) as a vector of length \(m\) over \(\GF{q}\). - Given a vector - vof length \(n\) over some field \(\GF{q^m}\), we can represent each entry as a vector of length \(m\), yielding an \(m \times n\) matrix over- sub_field. In case- sub_fieldis not given, we take the prime subfield \(\GF{p}\) of \(\GF{q^m}\).- INPUT: - v– a vector over some field \(\GF{q^m}\)
- sub_field– (default:- None) a sub field of \(\GF{q^m}\); if not specified, it is the prime subfield \(\GF{p}\) of \(\GF{q^m}\)
- basis– (default:- None) a basis of \(\GF{q^m}\) as a vector space over- sub_field. If not specified, the default basis is \(1,\beta,\ldots,\beta^{m-1}\) where \(\beta\) is the generator of \(\GF{q^m}\) given by SageMath.
 - EXAMPLES: - sage: from sage.coding.linear_rank_metric import to_matrix_representation sage: x = GF(64).gen() sage: a = vector(GF(64), (x + 1, x + 1, 1)) sage: to_matrix_representation(a, GF(4)) [1 1 1] [1 1 0] [0 0 0] sage: m = Matrix(GF(4), [[1, 1, 1], [1, 1, 0], [0, 0, 0]]) sage: to_matrix_representation(m) Traceback (most recent call last): ... TypeError: Input must be a vector - >>> from sage.all import * >>> from sage.coding.linear_rank_metric import to_matrix_representation >>> x = GF(Integer(64)).gen() >>> a = vector(GF(Integer(64)), (x + Integer(1), x + Integer(1), Integer(1))) >>> to_matrix_representation(a, GF(Integer(4))) [1 1 1] [1 1 0] [0 0 0] >>> m = Matrix(GF(Integer(4)), [[Integer(1), Integer(1), Integer(1)], [Integer(1), Integer(1), Integer(0)], [Integer(0), Integer(0), Integer(0)]]) >>> to_matrix_representation(m) Traceback (most recent call last): ... TypeError: Input must be a vector