def coset_leader(C, v):
"""
The vector v represents a received word, so should
be in the same ambient space V as C. Returns an
element of the syndrome of v of lowest weight.
EXAMPLES:
sage: C = HammingCode(2,GF(3)); C
Linear code of length 4, dimension 2 over Finite Field of size 3
sage: V = VectorSpace(GF(3), 4)
sage: v = V([0, 2, 0, 1])
sage: from sage.coding.decoder import coset_leader
sage: coset_leader(C, v)
((0, 0, 1, 0), 1)
sage: coset_leader(C, v)[0]-v in C
True
"""
coset = [[c+v, hamming_weight(c+v)] for c in C]
wts = [x[1] for x in coset]
min_wt = min(wts)
s = C[0] # initializing
w = hamming_weight(v) # initializing
for x in coset:
if x[1]==min_wt:
w = x[1]
s = x[0]
break
return s,w
def coset_leader(C, v):
"""
The vector v represents a received word, so should
be in the same ambient space V as C. Returns an
element of the syndrome of v of lowest weight.
EXAMPLES:
sage: C = HammingCode(2,GF(3)); C
Linear code of length 4, dimension 2 over Finite Field of size 3
sage: V = VectorSpace(GF(3), 4)
sage: v = V([0, 2, 0, 1])
sage: from sage.coding.decoder import coset_leader
sage: coset_leader(C, v)
((0, 0, 1, 0), 1)
sage: coset_leader(C, v)[0]-v in C
True
"""
coset = [[c + v, hamming_weight(c + v)] for c in C]
wts = [x[1] for x in coset]
min_wt = min(wts)
s = C[0] # initializing
w = hamming_weight(v) # initializing
for x in coset:
if x[1] == min_wt:
w = x[1]
s = x[0]
break
return s, w
def decode(C, v, algorithm="syndrome"):
"""
The vector v represents a received word, so should
be in the same ambient space V as C. Returns an
element in C which is closest to v in the Hamming
metric.
Methods implemented include "nearest neighbor" (essentially
a brute force search) and "syndrome".
EXAMPLES:
sage: C = HammingCode(2,GF(3))
sage: V = VectorSpace(GF(3), 4)
sage: v = V([0, 2, 0, 1])
sage: v in C
False
sage: from sage.coding.decoder import decode
sage: c = decode(C, v);c
(0, 2, 2, 1)
sage: c in C
True
sage: c = decode(C, v, algorithm="nearest neighbor");c
(0, 2, 2, 1)
sage: C = HammingCode(3,GF(3)); C
Linear code of length 13, dimension 10 over Finite Field of size 3
sage: V = VectorSpace(GF(3), 13)
sage: v = V([2]+[0]*12)
sage: decode(C, v) # long time (9s on sage.math, 2011)
(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
"""
V = C.ambient_space()
if not(type(v)==list):
v = v.list()
v = V(v)
if algorithm=="nearest neighbor":
diffs = [[c-v,hamming_weight(c-v)] for c in C]
diffs.sort(lambda x,y: x[1]-y[1])
return diffs[0][0]+v
if algorithm=="syndrome":
return -V(syndrome(C, v)[0])+v
def decode(C, v, algorithm="syndrome"):
"""
The vector v represents a received word, so should
be in the same ambient space V as C. Returns an
element in C which is closest to v in the Hamming
metric.
Methods implemented include "nearest neighbor" (essentially
a brute force search) and "syndrome".
EXAMPLES:
sage: C = HammingCode(2,GF(3))
sage: V = VectorSpace(GF(3), 4)
sage: v = V([0, 2, 0, 1])
sage: v in C
False
sage: from sage.coding.decoder import decode
sage: c = decode(C, v);c
(0, 2, 2, 1)
sage: c in C
True
sage: c = decode(C, v, algorithm="nearest neighbor");c
(0, 2, 2, 1)
sage: C = HammingCode(3,GF(3)); C
Linear code of length 13, dimension 10 over Finite Field of size 3
sage: V = VectorSpace(GF(3), 13)
sage: v = V([2]+[0]*12)
sage: decode(C, v) # long time (9s on sage.math, 2011)
(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
"""
V = C.ambient_space()
if not (type(v) == list):
v = v.list()
v = V(v)
if algorithm == "nearest neighbor":
diffs = [[c - v, hamming_weight(c - v)] for c in C]
diffs.sort(lambda x, y: x[1] - y[1])
return diffs[0][0] + v
if algorithm == "syndrome":
return -V(syndrome(C, v)[0]) + v
def syndrome(C, v):
"""
The vector v represents a received word, so should
be in the same ambient space V as C. Returns the
elements in V (including v) which belong to the
syndrome of v (ie, the coset v+C, sorted by weight).
EXAMPLES:
sage: C = HammingCode(2,GF(3)); C
Linear code of length 4, dimension 2 over Finite Field of size 3
sage: V = VectorSpace(GF(3), 4)
sage: v = V([0, 2, 0, 1])
sage: from sage.coding.decoder import syndrome
sage: syndrome(C, v)
[(0, 0, 1, 0), (0, 2, 0, 1), (2, 0, 0, 2), (1, 1, 0, 0), (2, 2, 2, 0), (1, 0, 2, 1), (0, 1, 2, 2), (1, 2, 1, 2), (2, 1, 1, 1)]
"""
V = C.ambient_space()
if not(type(v)==list):
v = v.list()
v = V(v)
coset = [[c+v,hamming_weight(c+v)] for c in C]
coset.sort(lambda x,y: x[1]-y[1])
return [x[0] for x in coset]
def syndrome(C, v):
"""
The vector v represents a received word, so should
be in the same ambient space V as C. Returns the
elements in V (including v) which belong to the
syndrome of v (ie, the coset v+C, sorted by weight).
EXAMPLES:
sage: C = HammingCode(2,GF(3)); C
Linear code of length 4, dimension 2 over Finite Field of size 3
sage: V = VectorSpace(GF(3), 4)
sage: v = V([0, 2, 0, 1])
sage: from sage.coding.decoder import syndrome
sage: syndrome(C, v)
[(0, 0, 1, 0), (0, 2, 0, 1), (2, 0, 0, 2), (1, 1, 0, 0), (2, 2, 2, 0), (1, 0, 2, 1), (0, 1, 2, 2), (1, 2, 1, 2), (2, 1, 1, 1)]
"""
V = C.ambient_space()
if not (type(v) == list):
v = v.list()
v = V(v)
coset = [[c + v, hamming_weight(c + v)] for c in C]
coset.sort(lambda x, y: x[1] - y[1])
return [x[0] for x in coset]
def is_block_design(self):
"""
Returns a pair True, pars if the incidence structure is a t-design,
for some t, where pars is the list of parameters [t, v, k, lmbda].
The largest possible t is returned, provided t=10.
EXAMPLES::
sage: BD = IncidenceStructure(range(7),[[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]])
sage: BD.is_block_design()
(True, [2, 7, 3, 1])
sage: BD.block_design_checker(2, 7, 3, 1)
True
sage: BD = WittDesign(9) # requires optional gap package 'design'
sage: BD.is_block_design() # requires optional gap package 'design'
(True, [2, 9, 3, 1])
sage: BD = WittDesign(12) # requires optional gap package 'design'
sage: BD.is_block_design() # requires optional gap package 'design'
(True, [5, 12, 6, 1])
sage: BD = AffineGeometryDesign(3, 1, GF(2))
sage: BD.is_block_design()
(True, [2, 8, 2, 2])
"""
from sage.combinat.designs.incidence_structures import coordinatewise_product
from sage.combinat.combinat import unordered_tuples, combinations
from sage.coding.linear_code import hamming_weight
A = self.incidence_matrix()
v = len(self.points())
b = len(self.blocks())
k = sum(A.columns()[0])
rowsA = A.rows()
VS = rowsA[0].parent()
r = sum(rowsA[0])
for i in range(b):
if not (sum(A.columns()[i]) == k):
return False
for i in range(v):
if not (sum(A.rows()[i]) == r):
return False
t_found_yet = False
lambdas = []
for t in range(2, min(v, 11)):
#print t
L1 = combinations(range(v), t)
L2 = [[rowsA[i] for i in L] for L in L1]
#print t,len(L2)
lmbda = hamming_weight(VS(coordinatewise_product(L2[0])))
lambdas.append(lmbda)
pars = [t, v, k, lmbda]
#print pars
for ell in L2:
a = hamming_weight(VS(coordinatewise_product(ell)))
if not (a == lmbda) or a == 0:
if not (t_found_yet):
pars = [t - 1, v, k, lambdas[t - 3]]
return False, pars
else:
#print pars, lambdas
pars = [t - 1, v, k, lambdas[t - 3]]
return True, pars
t_found_yet = True
pars = [t - 1, v, k, lambdas[t - 3]]
return True, pars
def is_block_design(self):
"""
Returns a pair True, pars if the incidence structure is a t-design,
for some t, where pars is the list of parameters [t, v, k, lmbda].
The largest possible t is returned, provided t=10.
EXAMPLES::
sage: BD = IncidenceStructure(range(7),[[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]])
sage: BD.is_block_design()
(True, [2, 7, 3, 1])
sage: BD.block_design_checker(2, 7, 3, 1)
True
sage: BD = WittDesign(9) # requires optional gap package 'design'
sage: BD.is_block_design() # requires optional gap package 'design'
(True, [2, 9, 3, 1])
sage: BD = WittDesign(12) # requires optional gap package 'design'
sage: BD.is_block_design() # requires optional gap package 'design'
(True, [5, 12, 6, 1])
sage: BD = AffineGeometryDesign(3, 1, GF(2))
sage: BD.is_block_design()
(True, [2, 8, 2, 2])
"""
from sage.combinat.designs.incidence_structures import coordinatewise_product
from sage.combinat.combinat import unordered_tuples, combinations
from sage.coding.linear_code import hamming_weight
A = self.incidence_matrix()
v = len(self.points())
b = len(self.blocks())
k = sum(A.columns()[0])
rowsA = A.rows()
VS = rowsA[0].parent()
r = sum(rowsA[0])
for i in range(b):
if not (sum(A.columns()[i]) == k):
return False
for i in range(v):
if not (sum(A.rows()[i]) == r):
return False
t_found_yet = False
lambdas = []
for t in range(2, min(v, 11)):
# print t
L1 = combinations(range(v), t)
L2 = [[rowsA[i] for i in L] for L in L1]
# print t,len(L2)
lmbda = hamming_weight(VS(coordinatewise_product(L2[0])))
lambdas.append(lmbda)
pars = [t, v, k, lmbda]
# print pars
for ell in L2:
a = hamming_weight(VS(coordinatewise_product(ell)))
if not (a == lmbda) or a == 0:
if not (t_found_yet):
pars = [t - 1, v, k, lambdas[t - 3]]
return False, pars
else:
# print pars, lambdas
pars = [t - 1, v, k, lambdas[t - 3]]
return True, pars
t_found_yet = True
pars = [t - 1, v, k, lambdas[t - 3]]
return True, pars