This is an answer showing why there is a memory issue while building the decoder for a bigger field $F$ of characteristic two and/or a higher degree of the involved polynomial $g\in \Bbb F_2[x]$.
To illustrate this, let us consider first the "smaller" case from the question:
F = GF(2^4)
R.<x> = F[]
g = x^2 + x + 1
L = [a for a in F if g(a)]
C = codes.GoppaCode(g, L)
D = C.decoder()
Then the object D has a lot of "hidden structure", for instance there is a dictionary D.__dict__, it has a lot of keys,
sage: for key in D.__dict__:
....: print(key)
....:
_maximum_error_weight
decoder_type
_code
_input_space
_connected_encoder_name
_build_lookup_table
_code_covering_radius
_code_minimum_distance
_decoder_type
_lookup_table
and for instance D.__dict__['_lookup_table'] is again a dictionary,
sage: print(type(D.__dict__['_lookup_table']))
<class 'dict'>
it has "a lot of keys", translated into "a lot of values". We can count:
sage: len(D.__dict__['_lookup_table'])
256
so there are $256$ keys, and some first few lookup translations are...
sage: D.__dict__['_lookup_table']
{(0, 0, 0, 0, 0, 0, 0, 0): (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),
(1, 0, 0, 0, 0, 0, 0, 0): (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),
(0, 1, 1, 0, 0, 0, 1, 1): (0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),
(1, 1, 1, 0, 1, 1, 1, 1): (0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),
(1, 1, 0, 1, 1, 1, 1, 0): (0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),
(0, 1, 1, 0, 0, 1, 0, 1): (0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0),
(1, 0, 0, 1, 0, 1, 1, 0): (0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0),
(0, 1, 1, 1, 0, 0, 0, 1): (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0),
(1, 1, 1, 0, 0, 0, 0, 1): (0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0),
and so on, totally $256$ key-value pairs.
If we try the same with a field $F=\Bbb F_{2^{10}}=$GF(2^10) and then further with a polynomial of degree $50$... well, there is no chance to build the corresponding structure. In such cases, maybe an ad-hoc solution should and may be given, try to see exactly what is needed from the decoder instance in the intended application.
Some further notes. One can look into the code, e.g. via C.decoder?? - and the code is mainly building...
if decoder_name in self._registered_decoders:
decClass = self._registered_decoders[decoder_name]
try:
return decClass(self, *args, **kwargs)
where self becomes the instance C, and the decoder_name is the default decoder,
sage: C._default_decoder_name
'Syndrome'
So we can build the corresponding class, and ask for some information...
sage: S = C._registered_decoders['Syndrome']
sage: S
<class 'sage.coding.linear_code.LinearCodeSyndromeDecoder'>
sage: S??
The double question mark gives...
Init signature: S(code, maximum_error_weight=None)
Docstring:
Constructs a decoder for Linear Codes based on syndrome lookup
table.
The decoding algorithm works as follows:
* First, a lookup table is built by computing the syndrome of every
error pattern of weight up to "maximum_error_weight".
* Then, whenever one tries to decode a word "r", the syndrome of
"r" is computed. The corresponding error pattern is recovered
from the pre-computed lookup table.
* Finally, the recovered error pattern is subtracted from "r" to
recover the original word.
"maximum_error_weight" need never exceed the covering radius of the
code, since there are then always lower-weight errors with the same
syndrome. If one sets "maximum_error_weight" to a value greater
than the covering radius, then the covering radius will be
determined while building the lookup-table. This lower value is
then returned if you query "decoding_radius" after construction.
If "maximum_error_weight" is left unspecified or set to a number at
least the covering radius of the code, this decoder is complete,
i.e. it decodes every vector in the ambient space.
Note:
Constructing the lookup table takes time exponential in the
length of the code and the size of the code's base field.
Afterwards, the individual decodings are fast.
INPUT:
* "code" -- A code associated to this decoder
* "maximum_error_weight" -- (default: "None") the maximum number of
errors to look for when building the table. An error is raised if
it is set greater than n-k, since this is an upper bound on the
covering radius on any linear code. If "maximum_error_weight" is
kept unspecified, it will be set to n - k, where n is the length
of "code" and k its dimension.
and a lot of further lines.
So a possibility to force some build of a decoder would be...
F = GF(2^10)
R.<x> = F[]
g = x^50 + x^3 + 1
L = [a for a in F if g(a)]
C = codes.GoppaCode(g, L)
D = C.decoder(maximum_error_weight=1)
Above, the option maximum_error_weight=1 may make the decoder useless for a lot of purposes, but this shows that the code is building D. We have in the above case:
sage: len(L)
1024
sage: len(D.__dict__['_lookup_table'])
1025
Using maximum_error_weight=2 instead, after a looonger wait we get a decoder with...
sage: len(D.__dict__['_lookup_table'])
524801
