r"""
Guava error-correcting code constructions.
This module only contains Guava wrappers (Guava is an optional GAP package).
AUTHOR:
-- David Joyner (2005-11-22, 2006-12-03): initial version
-- Nick Alexander (2006-12-10): factor GUAVA code to guava.py
-- David Joyner (2007-05): removed Golay codes, toric and trivial
codes and placed them in code_constructions;
renamed RandomLinearCode->RandomLinearCodeGuava
-- David Joyner (2008-03): removed QR, XQR, cyclic and ReedSolomon codes
-- " (2009-05): added "optional package" comments, fixed some docstrings to
to be sphinx compatible
"""
from sage.interfaces.all import gap
from sage.misc.randstate import current_randstate
from sage.misc.preparser import *
from sage.matrix.matrix_space import MatrixSpace
from sage.rings.finite_rings.constructor import FiniteField as GF
from sage.interfaces.gap import gfq_gap_to_sage
from sage.groups.perm_gps.permgroup import *
from linear_code import *
def BinaryReedMullerCode(r,k):
r"""
The binary 'Reed-Muller code' with dimension k and
order r is a code with length `2^k` and minimum distance `2^k-r`
(see for example, section 1.10 in [HP]_). By definition, the
`r^{th}` order binary Reed-Muller code of length `n=2^m`, for
`0 \leq r \leq m`, is the set of all vectors `(f(p)\ |\ p \\in GF(2)^m)`,
where `f` is a multivariate polynomial of degree at most `r`
in `m` variables.
INPUT:
r, k -- positive integers with `2^k>r`.
OUTPUT:
Returns the binary 'Reed-Muller code' with dimension k and order r.
EXAMPLE::
sage: C = BinaryReedMullerCode(2,4); C # requires optional package
Linear code of length 16, dimension 11 over Finite Field of size 2
sage: C.minimum_distance() # requires optional package
4
sage: C.gen_mat() # requires optional package
[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]
[0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1]
[0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1]
[0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1]
[0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1]
[0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1]
[0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1]
[0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1]
[0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1]
AUTHOR: David Joyner (11-2005)
"""
F = GF(2)
gap.eval("C:=ReedMullerCode("+str(r)+", "+str(k)+")")
gap.eval("G:=GeneratorMat(C)")
k = int(gap.eval("Length(G)"))
n = int(gap.eval("Length(G[1])"))
G = [[gfq_gap_to_sage(gap.eval("G["+str(i)+"]["+str(j)+"]"),F) for j in range(1,n+1)] for i in range(1,k+1)]
MS = MatrixSpace(F,k,n)
return LinearCode(MS(G))
def QuasiQuadraticResidueCode(p):
r"""
A (binary) quasi-quadratic residue code (or QQR code), as defined by
Proposition 2.2 in [BM]_, has a generator matrix in the block form `G=(Q,N)`.
Here `Q` is a `p \times p` circulant matrix whose top row
is `(0,x_1,...,x_{p-1})`, where `x_i=1` if and only if `i`
is a quadratic residue `\mod p`, and `N` is a `p \times p` circulant
matrix whose top row is `(0,y_1,...,y_{p-1})`, where `x_i+y_i=1` for all `i`.
INPUT:
p -- a prime >2.
OUTPUT:
Returns a QQR code of length 2p.
EXAMPLES::
sage: C = QuasiQuadraticResidueCode(11); C # requires optional package
Linear code of length 22, dimension 11 over Finite Field of size 2
REFERENCES:
..[BM] Bazzi and Mitter, {\it Some constructions of codes from group actions}, (preprint
March 2003, available on Mitter's MIT website).
..[J] D. Joyner, {\it On quadratic residue codes and hyperelliptic curves},
(preprint 2006)
These are self-orthogonal in general and self-dual when $p \\equiv 3 \\pmod 4$.
AUTHOR: David Joyner (11-2005)
"""
F = GF(2)
gap.eval("C:=QQRCode("+str(p)+")")
gap.eval("G:=GeneratorMat(C)")
k = int(gap.eval("Length(G)"))
n = int(gap.eval("Length(G[1])"))
G = [[gfq_gap_to_sage(gap.eval("G[%s][%s]" % (i,j)),F) for j in range(1,n+1)] for i in range(1,k+1)]
MS = MatrixSpace(F,k,n)
return LinearCode(MS(G))
def RandomLinearCodeGuava(n,k,F):
r"""
The method used is to first construct a `k \times n` matrix of the block
form `(I,A)`, where `I` is a `k \times k` identity matrix and `A` is a
`k \times (n-k)` matrix constructed using random elements of `F`. Then
the columns are permuted using a randomly selected element of the symmetric
group `S_n`.
INPUT:
Integers `n,k`, with `n>k>1`.
OUTPUT:
Returns a "random" linear code with length n, dimension k over field F.
EXAMPLES::
sage: C = RandomLinearCodeGuava(30,15,GF(2)); C # requires optional package
Linear code of length 30, dimension 15 over Finite Field of size 2
sage: C = RandomLinearCodeGuava(10,5,GF(4,'a')); C # requires optional package
Linear code of length 10, dimension 5 over Finite Field in a of size 2^2
AUTHOR: David Joyner (11-2005)
"""
current_randstate().set_seed_gap()
q = F.order()
gap.eval("C:=RandomLinearCode("+str(n)+","+str(k)+", GF("+str(q)+"))")
gap.eval("G:=GeneratorMat(C)")
k = int(gap.eval("Length(G)"))
n = int(gap.eval("Length(G[1])"))
G = [[gfq_gap_to_sage(gap.eval("G[%s][%s]" % (i,j)),F) for j in range(1,n+1)] for i in range(1,k+1)]
MS = MatrixSpace(F,k,n)
return LinearCode(MS(G))
