Subfield subcode

Let \(C\) be a \([n, k]\) code over \(\GF{q^t}\). Let \(Cs = \{c \in C | \forall i, c_i \in \GF{q}\}\), \(c_i\) being the \(i\)-th coordinate of \(c\).

\(Cs\) is called the subfield subcode of \(C\) over \(\GF{q}\)

class sage.coding.subfield_subcode.SubfieldSubcode(original_code, subfield, embedding=None)

Bases: sage.coding.linear_code.AbstractLinearCode

Representation of a subfield subcode.

INPUT:

  • original_code – the code self comes from.

  • subfield – the base field of self.

  • embedding – (default: None) an homomorphism from subfield to original_code’s base field. If None is provided, it will default to the first homomorphism of the list of homomorphisms Sage can build.

EXAMPLES:

sage: C = codes.random_linear_code(GF(16, 'aa'), 7, 3)
sage: codes.SubfieldSubcode(C, GF(4, 'a'))
Subfield subcode of [7, 3] linear code over GF(16) down to GF(4)
dimension()

Returns the dimension of self.

EXAMPLES:

sage: C = codes.GeneralizedReedSolomonCode(GF(16, 'aa').list()[:13], 5)
sage: Cs = codes.SubfieldSubcode(C, GF(4, 'a'))
sage: Cs.dimension()
3
dimension_lower_bound()

Returns a lower bound for the dimension of self.

EXAMPLES:

sage: C = codes.random_linear_code(GF(16, 'aa'), 7, 3)
sage: Cs = codes.SubfieldSubcode(C, GF(4, 'a'))
sage: Cs.dimension_lower_bound()
-1
dimension_upper_bound()

Returns an upper bound for the dimension of self.

EXAMPLES:

sage: C = codes.random_linear_code(GF(16, 'aa'), 7, 3)
sage: Cs = codes.SubfieldSubcode(C, GF(4, 'a'))
sage: Cs.dimension_upper_bound()
3
embedding()

Returns the field embedding between the base field of self and the base field of its original code.

EXAMPLES:

sage: C = codes.random_linear_code(GF(16, 'aa'), 7, 3)
sage: Cs = codes.SubfieldSubcode(C, GF(4, 'a'))
sage: Cs.embedding()
Ring morphism:
  From: Finite Field in a of size 2^2
  To:   Finite Field in aa of size 2^4
  Defn: a |--> aa^2 + aa
original_code()

Returns the original code of self.

EXAMPLES:

sage: C = codes.random_linear_code(GF(16, 'aa'), 7, 3)
sage: Cs = codes.SubfieldSubcode(C, GF(4, 'a'))
sage: Cs.original_code()
[7, 3] linear code over GF(16)
parity_check_matrix()

Returns a parity check matrix of self.

EXAMPLES:

sage: C = codes.GeneralizedReedSolomonCode(GF(16, 'aa').list()[:13], 5)
sage: Cs = codes.SubfieldSubcode(C, GF(4, 'a'))
sage: Cs.parity_check_matrix()
[    1     0     0     0     0     0     0     0     0     0     1 a + 1 a + 1]
[    0     1     0     0     0     0     0     0     0     0 a + 1     0     a]
[    0     0     1     0     0     0     0     0     0     0 a + 1     a     0]
[    0     0     0     1     0     0     0     0     0     0     0 a + 1     a]
[    0     0     0     0     1     0     0     0     0     0 a + 1     1 a + 1]
[    0     0     0     0     0     1     0     0     0     0     1     1     1]
[    0     0     0     0     0     0     1     0     0     0     a     a     1]
[    0     0     0     0     0     0     0     1     0     0     a     1     a]
[    0     0     0     0     0     0     0     0     1     0 a + 1 a + 1     1]
[    0     0     0     0     0     0     0     0     0     1     a     0 a + 1]
class sage.coding.subfield_subcode.SubfieldSubcodeOriginalCodeDecoder(code, original_decoder=None, **kwargs)

Bases: sage.coding.decoder.Decoder

Decoder decoding through a decoder over the original code of code.

INPUT:

  • code – The associated code of this decoder

  • original_decoder – (default: None) The decoder that will be used over the original code. It has to be a decoder object over the original code. If it is set to None, the default decoder over the original code will be used.

  • **kwargs – All extra arguments are forwarded to original code’s decoder

EXAMPLES:

sage: C = codes.GeneralizedReedSolomonCode(GF(16, 'aa').list()[:13], 5)
sage: Cs = codes.SubfieldSubcode(C, GF(4, 'a'))
sage: codes.decoders.SubfieldSubcodeOriginalCodeDecoder(Cs)
Decoder of Subfield subcode of [13, 5, 9] Reed-Solomon Code over GF(16) down to GF(4) through Gao decoder for [13, 5, 9] Reed-Solomon Code over GF(16)
decode_to_code(y)

Return an error-corrected codeword from y.

EXAMPLES:

sage: C = codes.GeneralizedReedSolomonCode(GF(16, 'aa').list()[:13], 5)
sage: Cs = codes.SubfieldSubcode(C, GF(4, 'a'))
sage: D = codes.decoders.SubfieldSubcodeOriginalCodeDecoder(Cs)
sage: Chan = channels.StaticErrorRateChannel(Cs.ambient_space(), D.decoding_radius())
sage: c = Cs.random_element()
sage: y = Chan(c)
sage: c == D.decode_to_code(y)
True
decoding_radius(**kwargs)

Returns maximal number of errors self can decode.

INPUT:

EXAMPLES:

sage: C = codes.GeneralizedReedSolomonCode(GF(16, 'aa').list()[:13], 5)
sage: Cs = codes.SubfieldSubcode(C, GF(4, 'a'))
sage: D = codes.decoders.SubfieldSubcodeOriginalCodeDecoder(Cs)
sage: D.decoding_radius()
4
original_decoder()

Returns the decoder over the original code that will be used to decode words of sage.coding.decoder.Decoder.code().

EXAMPLES:

sage: C = codes.GeneralizedReedSolomonCode(GF(16, 'aa').list()[:13], 5)
sage: Cs = codes.SubfieldSubcode(C, GF(4, 'a'))
sage: D = codes.decoders.SubfieldSubcodeOriginalCodeDecoder(Cs)
sage: D.original_decoder()
Gao decoder for [13, 5, 9] Reed-Solomon Code over GF(16)