Access functions to online databases for coding theory¶

sage.coding.databases.best_linear_code_in_codetables_dot_de(n, k, F, verbose=False)[source]¶

Return the best linear code and its construction as per the web database http://www.codetables.de/

INPUT:

n – integer; the length of the code
k – integer; the dimension of the code
F – finite field, of order 2, 3, 4, 5, 7, 8, or 9
verbose – boolean (default: False)

OUTPUT: an unparsed text explaining the construction of the code

EXAMPLES:

Sage

sage: L = codes.databases.best_linear_code_in_codetables_dot_de(72, 36, GF(2))    # optional - internet
sage: print(L)                                                                    # optional - internet
Construction of a linear code
[72,36,15] over GF(2):
[1]:  [73, 36, 16] Cyclic Linear Code over GF(2)
     CyclicCode of length 73 with generating polynomial x^37 + x^36 + x^34 +
x^33 + x^32 + x^27 + x^25 + x^24 + x^22 + x^21 + x^19 + x^18 + x^15 + x^11 +
x^10 + x^8 + x^7 + x^5 + x^3 + 1
[2]:  [72, 36, 15] Linear Code over GF(2)
     Puncturing of [1] at 1

last modified: 2002-03-20

Python

>>> from sage.all import *
>>> L = codes.databases.best_linear_code_in_codetables_dot_de(Integer(72), Integer(36), GF(Integer(2)))    # optional - internet
>>> print(L)                                                                    # optional - internet
Construction of a linear code
[72,36,15] over GF(2):
[1]:  [73, 36, 16] Cyclic Linear Code over GF(2)
     CyclicCode of length 73 with generating polynomial x^37 + x^36 + x^34 +
x^33 + x^32 + x^27 + x^25 + x^24 + x^22 + x^21 + x^19 + x^18 + x^15 + x^11 +
x^10 + x^8 + x^7 + x^5 + x^3 + 1
[2]:  [72, 36, 15] Linear Code over GF(2)
     Puncturing of [1] at 1
<BLANKLINE>
last modified: 2002-03-20

Sage Live

L = codes.databases.best_linear_code_in_codetables_dot_de(72, 36, GF(2))    # optional - internet
print(L)                                                                    # optional - internet

This function raises an IOError if an error occurs downloading data or parsing it. It raises a ValueError if the q input is invalid.

AUTHORS:

Steven Sivek (2005-11-14)
David Joyner (2008-03)

sage.coding.databases.best_linear_code_in_guava(n, k, F)[source]¶

Return the linear code of length n, dimension k over field F with the maximal minimum distance which is known to the GAP package GUAVA.

The function uses the tables described in bounds_on_minimum_distance_in_guava() to construct this code. This requires the optional GAP package GUAVA.

INPUT:

n – the length of the code to look up
k – the dimension of the code to look up
F – the base field of the code to look up

OUTPUT: a LinearCode which is a best linear code of the given parameters known to GUAVA

EXAMPLES:

Sage

sage: codes.databases.best_linear_code_in_guava(10,5,GF(2))    # long time; optional - gap_package_guava
[10, 5] linear code over GF(2)
sage: libgap.LoadPackage('guava')                              # long time; optional - gap_package_guava
...
sage: libgap.BestKnownLinearCode(10,5,libgap.GF(2))            # long time; optional - gap_package_guava
a linear [10,5,4]2..4 shortened code

Python

>>> from sage.all import *
>>> codes.databases.best_linear_code_in_guava(Integer(10),Integer(5),GF(Integer(2)))    # long time; optional - gap_package_guava
[10, 5] linear code over GF(2)
>>> libgap.LoadPackage('guava')                              # long time; optional - gap_package_guava
...
>>> libgap.BestKnownLinearCode(Integer(10),Integer(5),libgap.GF(Integer(2)))            # long time; optional - gap_package_guava
a linear [10,5,4]2..4 shortened code

Sage Live

codes.databases.best_linear_code_in_guava(10,5,GF(2))    # long time; optional - gap_package_guava
libgap.LoadPackage('guava')                              # long time; optional - gap_package_guava
libgap.BestKnownLinearCode(10,5,libgap.GF(2))            # long time; optional - gap_package_guava

This means that the best possible binary linear code of length 10 and dimension 5 is a code with minimum distance 4 and covering radius s somewhere between 2 and 4. Use bounds_on_minimum_distance_in_guava(10,5,GF(2)) for further details.

sage.coding.databases.bounds_on_minimum_distance_in_guava(n, k, F)[source]¶

Compute a lower and upper bound on the greatest minimum distance of a \([n,k]\) linear code over the field F.

This function requires the optional GAP package GUAVA.

The function returns a GAP record with the two bounds and an explanation for each bound. The method Display can be used to show the explanations.

The values for the lower and upper bound are obtained from a table constructed by Cen Tjhai for GUAVA, derived from the table of Brouwer. See http://www.codetables.de/ for the most recent data. These tables contain lower and upper bounds for \(q=2\) (when n <= 257), \(q=3\) (when n <= 243), \(q=4\) (n <= 256). (Current as of 11 May 2006.) For codes over other fields and for larger word lengths, trivial bounds are used.

INPUT:

n – the length of the code to look up
k – the dimension of the code to look up
F – the base field of the code to look up

OUTPUT: a GAP record object. See below for an example

EXAMPLES:

Sage

sage: gap_rec = codes.databases.bounds_on_minimum_distance_in_guava(10,5,GF(2))  # optional - gap_package_guava
sage: gap_rec.Display()                                                          # optional - gap_package_guava
rec(
  construction := [ <Operation "ShortenedCode">,
    [ [ <Operation "UUVCode">,
      [ [ <Operation "DualCode">,
      [ [ <Operation "RepetitionCode">, [ 8, 2 ] ] ] ],
      [ <Operation "UUVCode">, [ [ <Operation "DualCode">,
        [ [ <Operation "RepetitionCode">, [ 4, 2 ] ] ] ],
        [ <Operation "RepetitionCode">, [ 4, 2 ] ] ] ] ] ],
    [ 1, 2, 3, 4, 5, 6 ] ] ],
  k := 5,
  lowerBound := 4,
  lowerBoundExplanation := ...
  n := 10,
  q := 2,
  references := rec(
       ),
  upperBound := 4,
  upperBoundExplanation := ... )

Python

>>> from sage.all import *
>>> gap_rec = codes.databases.bounds_on_minimum_distance_in_guava(Integer(10),Integer(5),GF(Integer(2)))  # optional - gap_package_guava
>>> gap_rec.Display()                                                          # optional - gap_package_guava
rec(
  construction := [ <Operation "ShortenedCode">,
    [ [ <Operation "UUVCode">,
      [ [ <Operation "DualCode">,
      [ [ <Operation "RepetitionCode">, [ 8, 2 ] ] ] ],
      [ <Operation "UUVCode">, [ [ <Operation "DualCode">,
        [ [ <Operation "RepetitionCode">, [ 4, 2 ] ] ] ],
        [ <Operation "RepetitionCode">, [ 4, 2 ] ] ] ] ] ],
    [ 1, 2, 3, 4, 5, 6 ] ] ],
  k := 5,
  lowerBound := 4,
  lowerBoundExplanation := ...
  n := 10,
  q := 2,
  references := rec(
       ),
  upperBound := 4,
  upperBoundExplanation := ... )

Sage Live

gap_rec = codes.databases.bounds_on_minimum_distance_in_guava(10,5,GF(2))  # optional - gap_package_guava
gap_rec.Display()                                                          # optional - gap_package_guava

sage.coding.databases.self_orthogonal_binary_codes(n, k, b=2, parent=None, BC=None, equal=False, in_test=None)[source]¶

Return a Python iterator which generates a complete set of representatives of all permutation equivalence classes of self-orthogonal binary linear codes of length in [1..n] and dimension in [1..k].

INPUT:

n – integer; maximal length
k – integer; maximal dimension
b – integer; requires that the generators all have weight divisible by b (if b=2, all self-orthogonal codes are generated, and if b=4, all doubly even codes are generated). Must be an even positive integer.
parent – used in recursion (default: None)
BC – used in recursion (default: None)
equal – if True, generates only [n, k] codes (default: False)
in_test – used in recursion (default: None)

EXAMPLES:

Generate all self-orthogonal codes of length up to 7 and dimension up to 3:

Sage

sage: # needs sage.groups
sage: for B in codes.databases.self_orthogonal_binary_codes(7,3):
....:    print(B)
[2, 1] linear code over GF(2)
[4, 2] linear code over GF(2)
[6, 3] linear code over GF(2)
[4, 1] linear code over GF(2)
[6, 2] linear code over GF(2)
[6, 2] linear code over GF(2)
[7, 3] linear code over GF(2)
[6, 1] linear code over GF(2)

Python

>>> from sage.all import *
>>> # needs sage.groups
>>> for B in codes.databases.self_orthogonal_binary_codes(Integer(7),Integer(3)):
...    print(B)
[2, 1] linear code over GF(2)
[4, 2] linear code over GF(2)
[6, 3] linear code over GF(2)
[4, 1] linear code over GF(2)
[6, 2] linear code over GF(2)
[6, 2] linear code over GF(2)
[7, 3] linear code over GF(2)
[6, 1] linear code over GF(2)

Sage Live

# needs sage.groups
for B in codes.databases.self_orthogonal_binary_codes(7,3):
   print(B)

Generate all doubly-even codes of length up to 7 and dimension up to 3:

Sage

sage: # needs sage.groups
sage: for B in codes.databases.self_orthogonal_binary_codes(7,3,4):
....:    print(B); print(B.generator_matrix())
[4, 1] linear code over GF(2)
[1 1 1 1]
[6, 2] linear code over GF(2)
[1 1 1 1 0 0]
[0 1 0 1 1 1]
[7, 3] linear code over GF(2)
[1 0 1 1 0 1 0]
[0 1 0 1 1 1 0]
[0 0 1 0 1 1 1]

Python

>>> from sage.all import *
>>> # needs sage.groups
>>> for B in codes.databases.self_orthogonal_binary_codes(Integer(7),Integer(3),Integer(4)):
...    print(B); print(B.generator_matrix())
[4, 1] linear code over GF(2)
[1 1 1 1]
[6, 2] linear code over GF(2)
[1 1 1 1 0 0]
[0 1 0 1 1 1]
[7, 3] linear code over GF(2)
[1 0 1 1 0 1 0]
[0 1 0 1 1 1 0]
[0 0 1 0 1 1 1]

Sage Live

# needs sage.groups
for B in codes.databases.self_orthogonal_binary_codes(7,3,4):
   print(B); print(B.generator_matrix())

Generate all doubly-even codes of length up to 7 and dimension up to 2:

Sage

sage: # needs sage.groups
sage: for B in codes.databases.self_orthogonal_binary_codes(7,2,4):
....:    print(B); print(B.generator_matrix())
[4, 1] linear code over GF(2)
[1 1 1 1]
[6, 2] linear code over GF(2)
[1 1 1 1 0 0]
[0 1 0 1 1 1]

Python

>>> from sage.all import *
>>> # needs sage.groups
>>> for B in codes.databases.self_orthogonal_binary_codes(Integer(7),Integer(2),Integer(4)):
...    print(B); print(B.generator_matrix())
[4, 1] linear code over GF(2)
[1 1 1 1]
[6, 2] linear code over GF(2)
[1 1 1 1 0 0]
[0 1 0 1 1 1]

Sage Live

# needs sage.groups
for B in codes.databases.self_orthogonal_binary_codes(7,2,4):
   print(B); print(B.generator_matrix())

Generate all self-orthogonal codes of length equal to 8 and dimension equal to 4:

Sage

sage: # needs sage.groups
sage: for B in codes.databases.self_orthogonal_binary_codes(8, 4, equal=True):
....:     print(B); print(B.generator_matrix())
[8, 4] linear code over GF(2)
[1 0 0 1 0 0 0 0]
[0 1 0 0 1 0 0 0]
[0 0 1 0 0 1 0 0]
[0 0 0 0 0 0 1 1]
[8, 4] linear code over GF(2)
[1 0 0 1 1 0 1 0]
[0 1 0 1 1 1 0 0]
[0 0 1 0 1 1 1 0]
[0 0 0 1 0 1 1 1]

Python

>>> from sage.all import *
>>> # needs sage.groups
>>> for B in codes.databases.self_orthogonal_binary_codes(Integer(8), Integer(4), equal=True):
...     print(B); print(B.generator_matrix())
[8, 4] linear code over GF(2)
[1 0 0 1 0 0 0 0]
[0 1 0 0 1 0 0 0]
[0 0 1 0 0 1 0 0]
[0 0 0 0 0 0 1 1]
[8, 4] linear code over GF(2)
[1 0 0 1 1 0 1 0]
[0 1 0 1 1 1 0 0]
[0 0 1 0 1 1 1 0]
[0 0 0 1 0 1 1 1]

Sage Live

# needs sage.groups
for B in codes.databases.self_orthogonal_binary_codes(8, 4, equal=True):
    print(B); print(B.generator_matrix())

Since all the codes will be self-orthogonal, b must be divisible by 2:

Sage

sage: list(codes.databases.self_orthogonal_binary_codes(8, 4, 1, equal=True))
Traceback (most recent call last):
...
ValueError: b (1) must be a positive even integer.

Python

>>> from sage.all import *
>>> list(codes.databases.self_orthogonal_binary_codes(Integer(8), Integer(4), Integer(1), equal=True))
Traceback (most recent call last):
...
ValueError: b (1) must be a positive even integer.

Sage Live

list(codes.databases.self_orthogonal_binary_codes(8, 4, 1, equal=True))