def cartan_matrix(self, subdivide=True):
"""
Return the Cartan matrix associated with ``self``. By default
the Cartan matrix is a subdivided block matrix showing the
reducibility but the subdivision can be suppressed with
the option ``subdivide = False``.
.. TODO::
Currently ``subdivide`` is currently ignored.
EXAMPLES::
sage: ct = CartanType("A2","B2")
sage: ct.cartan_matrix()
[ 2 -1| 0 0]
[-1 2| 0 0]
[-----+-----]
[ 0 0| 2 -1]
[ 0 0|-2 2]
sage: ct.cartan_matrix(subdivide=False)
[ 2 -1 0 0]
[-1 2 0 0]
[ 0 0 2 -1]
[ 0 0 -2 2]
"""
from sage.combinat.root_system.cartan_matrix import CartanMatrix
return CartanMatrix(block_diagonal_matrix([t.cartan_matrix() for t in self._types], subdivide=subdivide),
cartan_type=self)
def cartan_matrix(self, subdivide=True):
"""
Return the Cartan matrix associated with ``self``. By default
the Cartan matrix is a subdivided block matrix showing the
reducibility but the subdivision can be suppressed with
the option ``subdivide = False``.
EXAMPLES::
sage: ct = CartanType("A2","B2")
sage: ct.cartan_matrix()
[ 2 -1| 0 0]
[-1 2| 0 0]
[-----+-----]
[ 0 0| 2 -1]
[ 0 0|-2 2]
sage: ct.cartan_matrix(subdivide=False)
[ 2 -1 0 0]
[-1 2 0 0]
[ 0 0 2 -1]
[ 0 0 -2 2]
"""
from sage.combinat.root_system.cartan_matrix import CartanMatrix
return CartanMatrix(block_diagonal_matrix(
[t.cartan_matrix() for t in self._types], subdivide=subdivide),
cartan_type=self)
def to_matrix(self, deg=None):
"""
The matrix representing this chain map.
If the degree ``deg`` is specified, return the matrix in that
degree; otherwise, return the (block) matrix for the whole
chain map.
INPUT:
- ``deg`` -- (optional, default ``None``) the degree
EXAMPLES::
sage: C = ChainComplex({0: identity_matrix(ZZ, 1)})
sage: D = ChainComplex({0: zero_matrix(ZZ, 1), 1: zero_matrix(ZZ, 1)})
sage: f = Hom(C,D)({0: identity_matrix(ZZ, 1), 1: zero_matrix(ZZ, 1)})
sage: f.to_matrix(0)
[1]
sage: f.to_matrix()
[1|0|]
[-+-+]
[0|0|]
[-+-+]
[0|0|]
"""
if deg is not None:
return self.in_degree(deg)
row = 0
col = 0
blocks = [self._matrix_dictionary[n]
for n in sorted(self._matrix_dictionary.keys())]
return block_diagonal_matrix(blocks)
def to_matrix(self, deg=None):
"""
The matrix representing this chain map.
If the degree ``deg`` is specified, return the matrix in that
degree; otherwise, return the (block) matrix for the whole
chain map.
INPUT:
- ``deg`` -- (optional, default ``None``) the degree
EXAMPLES::
sage: C = ChainComplex({0: identity_matrix(ZZ, 1)})
sage: D = ChainComplex({0: zero_matrix(ZZ, 1), 1: zero_matrix(ZZ, 1)})
sage: f = Hom(C,D)({0: identity_matrix(ZZ, 1), 1: zero_matrix(ZZ, 1)})
sage: f.to_matrix(0)
[1]
sage: f.to_matrix()
[1|0|]
[-+-+]
[0|0|]
[-+-+]
[0|0|]
"""
if deg is not None:
return self.in_degree(deg)
row = 0
col = 0
blocks = [
self._matrix_dictionary[n]
for n in sorted(self._matrix_dictionary.keys())
]
return block_diagonal_matrix(blocks)
def lift(A, N):
r"""
Lift a matrix A from SL_m(Z/NZ) to SL_m(Z).
Follows Shimura, Lemma 1.38, p21.
"""
assert A.is_square()
assert A.determinant() != 0
m = A.nrows()
if m == 1:
return identity_matrix(1)
D, U, V = A.smith_form()
if U.determinant() == -1:
U = matrix(2, 2, [-1, 0, 0, 1]) * U
if V.determinant() == -1:
V = V * matrix(2, 2, [-1, 0, 0, 1])
D = U * A * V
assert U.determinant() == 1
assert V.determinant() == 1
a = [D[i, i] for i in range(m)]
b = prod(a[1:])
W = identity_matrix(m)
W[0, 0] = b
W[1, 0] = b - 1
W[0, 1] = 1
X = identity_matrix(m)
X[0, 1] = -a[1]
Ap = D.parent()(D)
Ap[0, 0] = 1
Ap[1, 0] = 1 - a[0]
Ap[1, 1] *= a[0]
assert (W * U * A * V * X).change_ring(Zmod(N)) == Ap.change_ring(Zmod(N))
Cp = diagonal_matrix(a[1:])
Cp[0, 0] *= a[0]
C = lift(Cp, N)
Cpp = block_diagonal_matrix(identity_matrix(1), C)
Cpp[1, 0] = 1 - a[0]
return (~U * ~W * Cpp * ~X * ~V).change_ring(ZZ)
def lift(A, N):
r"""
Lift a matrix A from SL_m(Z/NZ) to SL_m(Z).
Follows Shimura, Lemma 1.38, p21.
"""
assert A.is_square()
assert A.determinant() != 0
m = A.nrows()
if m == 1:
return identity_matrix(1)
D, U, V = A.smith_form()
if U.determinant() == -1 :
U = matrix(2,2,[-1,0,0,1])* U
if V.determinant() == -1 :
V = V *matrix(2,2,[-1,0,0,1])
D = U*A*V
assert U.determinant() == 1
assert V.determinant() == 1
a = [ D[i, i] for i in range(m) ]
b = prod(a[1:])
W = identity_matrix(m)
W[0, 0] = b
W[1, 0] = b-1
W[0, 1] = 1
X = identity_matrix(m)
X[0, 1] = -a[1]
Ap = D.parent()(D)
Ap[0, 0] = 1
Ap[1, 0] = 1-a[0]
Ap[1, 1] *= a[0]
assert (W*U*A*V*X).change_ring(Zmod(N)) == Ap.change_ring(Zmod(N))
Cp = diagonal_matrix(a[1:])
Cp[0, 0] *= a[0]
C = lift(Cp, N)
Cpp = block_diagonal_matrix(identity_matrix(1), C)
Cpp[1, 0] = 1-a[0]
return (~U * ~W * Cpp * ~X * ~V).change_ring(ZZ)
def cartan_matrix(self, subdivide=True):
"""
Returns the Cartan matrix associated with self. By default
the Cartan matrix is a subdivided block matrix showing the
reducibility but the subdivision can be suppressed with
the option subdivide=False.
EXAMPLES::
sage: ct = CartanType("A2","B2")
sage: ct.cartan_matrix()
[ 2 -1| 0 0]
[-1 2| 0 0]
[-----+-----]
[ 0 0| 2 -1]
[ 0 0|-2 2]
sage: ct.cartan_matrix(subdivide=False)
[ 2 -1 0 0]
[-1 2 0 0]
[ 0 0 2 -1]
[ 0 0 -2 2]
"""
return block_diagonal_matrix([t.cartan_matrix() for t in self._types], subdivide=subdivide)
def cartan_matrix(self, subdivide=True):
"""
Returns the Cartan matrix associated with self. By default
the Cartan matrix is a subdivided block matrix showing the
reducibility but the subdivision can be suppressed with
the option subdivide=False.
EXAMPLES::
sage: ct = CartanType("A2","B2")
sage: ct.cartan_matrix()
[ 2 -1| 0 0]
[-1 2| 0 0]
[-----+-----]
[ 0 0| 2 -1]
[ 0 0|-2 2]
sage: ct.cartan_matrix(subdivide=False)
[ 2 -1 0 0]
[-1 2 0 0]
[ 0 0 2 -1]
[ 0 0 -2 2]
"""
return block_diagonal_matrix([t.cartan_matrix() for t in self._types], subdivide=subdivide)
def rshcd_from_close_prime_powers(n):
r"""
Return a `(n^2,1)`-RSHCD when `n-1` and `n+1` are odd prime powers and `n=0\pmod{4}`.
The construction implemented here appears in Theorem 4.3 from [GS70]_.
Note that the authors of [SWW72]_ claim in Corollary 5.12 (page 342) to have
proved the same result without the `n=0\pmod{4}` restriction with a *very*
similar construction. So far, however, I (Nathann Cohen) have not been able
to make it work.
INPUT:
- ``n`` -- an integer congruent to `0\pmod{4}`
.. SEEALSO::
:func:`regular_symmetric_hadamard_matrix_with_constant_diagonal`
EXAMPLES::
sage: from sage.combinat.matrices.hadamard_matrix import rshcd_from_close_prime_powers
sage: rshcd_from_close_prime_powers(4)
[-1 -1 1 -1 1 -1 -1 1 -1 1 -1 -1 1 -1 1 -1]
[-1 -1 1 1 -1 -1 -1 -1 -1 1 1 -1 -1 1 -1 1]
[ 1 1 -1 1 1 -1 -1 -1 -1 -1 1 -1 -1 -1 1 -1]
[-1 1 1 -1 1 1 -1 -1 -1 -1 -1 1 -1 -1 -1 1]
[ 1 -1 1 1 -1 1 1 -1 -1 -1 -1 -1 1 -1 -1 -1]
[-1 -1 -1 1 1 -1 1 1 -1 -1 -1 1 -1 1 -1 -1]
[-1 -1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 1 -1 -1]
[ 1 -1 -1 -1 -1 1 -1 -1 -1 1 -1 1 -1 1 1 -1]
[-1 -1 -1 -1 -1 -1 1 -1 -1 -1 1 1 1 -1 1 1]
[ 1 1 -1 -1 -1 -1 -1 1 -1 -1 -1 -1 1 1 -1 1]
[-1 1 1 -1 -1 -1 1 -1 1 -1 -1 -1 -1 1 1 -1]
[-1 -1 -1 1 -1 1 -1 1 1 -1 -1 -1 -1 -1 1 1]
[ 1 -1 -1 -1 1 -1 1 -1 1 1 -1 -1 -1 -1 -1 1]
[-1 1 -1 -1 -1 1 1 1 -1 1 1 -1 -1 -1 -1 -1]
[ 1 -1 1 -1 -1 -1 -1 1 1 -1 1 1 -1 -1 -1 -1]
[-1 1 -1 1 -1 -1 -1 -1 1 1 -1 1 1 -1 -1 -1]
REFERENCE:
.. [SWW72] A Street, W. Wallis, J. Wallis,
Combinatorics: Room squares, sum-free sets, Hadamard matrices.
Lecture notes in Mathematics 292 (1972).
"""
if n%4:
raise ValueError("n(={}) must be congruent to 0 mod 4")
a,b = sorted([n-1,n+1],key=lambda x:-x%4)
Sa = _helper_payley_matrix(a)
Sb = _helper_payley_matrix(b)
U = matrix(a,[[int(i+j == a-1) for i in range(a)] for j in range(a)])
K = (U*Sa).tensor_product(Sb) + U.tensor_product(J(b)-I(b)) - J(a).tensor_product(I(b))
F = lambda x:diagonal_matrix([-(-1)**i for i in range(x)])
G = block_diagonal_matrix([J(1),I(a).tensor_product(F(b))])
e = matrix(a*b,[1]*(a*b))
H = block_matrix(2,[-J(1),e.transpose(),e,K])
HH = G*H*G
assert len(set(map(sum,HH))) == 1
assert HH**2 == n**2*I(n**2)
return HH
def rshcd_from_close_prime_powers(n):
r"""
Return a `(n^2,1)`-RSHCD when `n-1` and `n+1` are odd prime powers and `n=0\pmod{4}`.
The construction implemented here appears in Theorem 4.3 from [GS70]_.
Note that the authors of [SWW72]_ claim in Corollary 5.12 (page 342) to have
proved the same result without the `n=0\pmod{4}` restriction with a *very*
similar construction. So far, however, I (Nathann Cohen) have not been able
to make it work.
INPUT:
- ``n`` -- an integer congruent to `0\pmod{4}`
.. SEEALSO::
:func:`regular_symmetric_hadamard_matrix_with_constant_diagonal`
EXAMPLES::
sage: from sage.combinat.matrices.hadamard_matrix import rshcd_from_close_prime_powers
sage: rshcd_from_close_prime_powers(4)
[-1 -1 1 -1 1 -1 -1 1 -1 1 -1 -1 1 -1 1 -1]
[-1 -1 1 1 -1 -1 -1 -1 -1 1 1 -1 -1 1 -1 1]
[ 1 1 -1 1 1 -1 -1 -1 -1 -1 1 -1 -1 -1 1 -1]
[-1 1 1 -1 1 1 -1 -1 -1 -1 -1 1 -1 -1 -1 1]
[ 1 -1 1 1 -1 1 1 -1 -1 -1 -1 -1 1 -1 -1 -1]
[-1 -1 -1 1 1 -1 1 1 -1 -1 -1 1 -1 1 -1 -1]
[-1 -1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 1 -1 -1]
[ 1 -1 -1 -1 -1 1 -1 -1 -1 1 -1 1 -1 1 1 -1]
[-1 -1 -1 -1 -1 -1 1 -1 -1 -1 1 1 1 -1 1 1]
[ 1 1 -1 -1 -1 -1 -1 1 -1 -1 -1 -1 1 1 -1 1]
[-1 1 1 -1 -1 -1 1 -1 1 -1 -1 -1 -1 1 1 -1]
[-1 -1 -1 1 -1 1 -1 1 1 -1 -1 -1 -1 -1 1 1]
[ 1 -1 -1 -1 1 -1 1 -1 1 1 -1 -1 -1 -1 -1 1]
[-1 1 -1 -1 -1 1 1 1 -1 1 1 -1 -1 -1 -1 -1]
[ 1 -1 1 -1 -1 -1 -1 1 1 -1 1 1 -1 -1 -1 -1]
[-1 1 -1 1 -1 -1 -1 -1 1 1 -1 1 1 -1 -1 -1]
REFERENCE:
.. [SWW72] A Street, W. Wallis, J. Wallis,
Combinatorics: Room squares, sum-free sets, Hadamard matrices.
Lecture notes in Mathematics 292 (1972).
"""
if n % 4:
raise ValueError("n(={}) must be congruent to 0 mod 4")
a, b = sorted([n - 1, n + 1], key=lambda x: -x % 4)
Sa = _helper_payley_matrix(a)
Sb = _helper_payley_matrix(b)
U = matrix(a, [[int(i + j == a - 1) for i in range(a)] for j in range(a)])
K = (U * Sa).tensor_product(Sb) + U.tensor_product(J(b) - I(b)) - J(
a).tensor_product(I(b))
F = lambda x: diagonal_matrix([-(-1)**i for i in range(x)])
G = block_diagonal_matrix([J(1), I(a).tensor_product(F(b))])
e = matrix(a * b, [1] * (a * b))
H = block_matrix(2, [-J(1), e.transpose(), e, K])
HH = G * H * G
assert len(set(map(sum, HH))) == 1
assert HH**2 == n**2 * I(n**2)
return HH
def self_dual_codes_binary(n):
r"""
Returns the dictionary of inequivalent sd codes of length n.
For n=4 even, returns the sd codes of a given length, up to (perm)
equivalence, the (perm) aut gp, and the type.
The number of inequiv "diagonal" sd binary codes in the database of
length n is ("diagonal" is defined by the conjecture above) is the
same as the restricted partition number of n, where only integers
from the set 1,4,6,8,... are allowed. This is the coefficient of
`x^n` in the series expansion
`(1-x)^{-1}\prod_{2^\infty (1-x^{2j})^{-1}}`. Typing the
command f = (1-x)(-1)\*prod([(1-x(2\*j))(-1) for j in range(2,18)])
into Sage, we obtain for the coeffs of `x^4`,
`x^6`, ... [1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 11, 11, 15, 15,
22, 22, 30, 30, 42, 42, 56, 56, 77, 77, 101, 101, 135, 135, 176,
176, 231] These numbers grow too slowly to account for all the sd
codes (see Huffman+Pless' Table 9.1, referenced above). In fact, in
Table 9.10 of [HP], the number B_n of inequivalent sd binary codes
of length n is given::
n 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
B_n 1 1 1 2 2 3 4 7 9 16 25 55 103 261 731
According to http://oeis.org/classic/A003179,
the next 2 entries are: 3295, 24147.
EXAMPLES::
sage: C = self_dual_codes_binary(10)
sage: C["10"]["0"]["code"] == C["10"]["0"]["code"].dual_code()
True
sage: C["10"]["1"]["code"] == C["10"]["1"]["code"].dual_code()
True
sage: len(C["10"].keys()) # number of inequiv sd codes of length 10
2
sage: C = self_dual_codes_binary(12)
sage: C["12"]["0"]["code"] == C["12"]["0"]["code"].dual_code()
True
sage: C["12"]["1"]["code"] == C["12"]["1"]["code"].dual_code()
True
sage: C["12"]["2"]["code"] == C["12"]["2"]["code"].dual_code()
True
"""
sd_codes = {}
if n == 4:
# this code is Type I
# [4,0]:
genmat = I2(n).augment(I2(n))
# G = PermutationGroup([ "(2,4)", "(1,2)(3,4)" ])
spectrum = [1, 0, 2, 0, 1]
sd_codes_4_0 = {"order autgp":8,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":"Unique."}
sd_codes["4"] = {"0":sd_codes_4_0}
return sd_codes
if n == 6:
# this is Type I
# [6,0]:
genmat = I2(n).augment(I2(n))
# G = PermutationGroup( ["(3,6)", "(2,3)(5,6)", "(1,2)(4,5)"] )
spectrum = [1, 0, 3, 0, 3, 0, 1]
sd_codes_6_0 = {"order autgp":48,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":"Unique"}
sd_codes["6"] = {"0":sd_codes_6_0}
return sd_codes
if n == 8:
# the first code is Type I, the second is Type II
# the second code is equiv to the extended Hamming [8,4,4] code.
# [8,0]:
genmat = I2(n).augment(I2(n))
# G = PermutationGroup( ["(4,8)", "(3,4)(7,8)", "(2,3)(6,7)", "(1,2)(5,6)"] )
spectrum = [1, 0, 4, 0, 6, 0, 4, 0, 1]
sd_codes_8_0 = {"order autgp":384,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":"Unique Type I of this length."}
# [8,1]:
genmat = I2(n).augment(matA(n)[4])
# G = PermutationGroup( ["(4,5)(6,7)", "(4,6)(5,7)", "(3,4)(7,8)",\
# "(2,3)(6,7)", "(1,2)(5,6)"] )
spectrum = [1, 0, 0, 0, 14, 0, 0, 0, 1]
sd_codes_8_1 = {"order autgp":1344,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"II","Comment":"Unique Type II of this length."}
sd_codes["8"] = {"0":sd_codes_8_0,"1":sd_codes_8_1}
return sd_codes
if n == 10:
# Both of these are Type I; one has a unique lowest weight codeword
# [10,0]:
genmat = I2(n).augment(I2(n))
# G = PermutationGroup( ["(5,10)", "(4,5)(9,10)", "(3,4)(8,9)",\
# "(2,3)(7,8)", "(1,2)(6,7)"] )
spectrum = [1, 0, 5, 0, 10, 0, 10, 0, 5, 0, 1]
sd_codes_10_0 = {"order autgp":3840,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":"No Type II of this length."}
# [10,1]:
genmat = I2(n).augment(block_diagonal_matrix([matA(n)[4],matId(n)[4]]))
# G = PermutationGroup( ["(5,10)", "(4,6)(7,8)", "(4,7)(6,8)", "(3,4)(8,9)",\
# "(2,3)(7,8)", "(1,2)(6,7)"] )
spectrum = [1, 0, 1, 0, 14, 0, 14, 0, 1, 0, 1]
sd_codes_10_1 = {"order autgp":2688,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":"Unique lowest weight nonzero codeword."}
sd_codes["10"] = {"0":sd_codes_10_0,"1":sd_codes_10_1}
return sd_codes
if n == 12:
# all of these are Type I
# [12,0]:
genmat = I2(n).augment(I2(n))
# G = PermutationGroup( ["(6,12)", "(5,6)(11,12)", "(4,5)(10,11)", "(3,4)(9,10)",\
# "(2,3)(8,9)", "(1,2)(7,8)"] )
spectrum = [1, 0, 6, 0, 15, 0, 20, 0, 15, 0, 6, 0, 1]
sd_codes_12_0 = {"order autgp":48080,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":"No Type II of this length."}
# [12,1]:
genmat = I2(n).augment(block_diagonal_matrix([matA(n)[4],matId(n)[4]]))
# G = PermutationGroup( ["(2,3)(4,7)", "(2,4)(3,7)", "(2,4,9)(3,7,8)", "(2,4,8,10)(3,9)",\
# "(1,2,4,7,8,10)(3,9)", "(2,4,8,10)(3,9)(6,12)", "(2,4,8,10)(3,9)(5,6,11,12)"] )
spectrum = [1, 0, 2, 0, 15, 0, 28, 0, 15, 0, 2, 0, 1]
sd_codes_12_1 = {"order autgp":10752,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":"Smallest automorphism group of these."}
# [12,2]:
genmat = I2(n).augment(matA(n)[6])
# G = PermutationGroup( ["(5,6)(11,12)", "(5,11)(6,12)", "(4,5)(10,11)", "(3,4)(9,10)",\
# "(2,3)(8,9)", "(1,2)(7,8)"] )
spectrum = [1, 0, 0, 0, 15, 0, 32, 0, 15, 0, 0, 0, 1]
sd_codes_12_2 = {"order autgp":23040,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":"Largest minimum distance of these."}
sd_codes["12"] = {"0":sd_codes_12_0,"1":sd_codes_12_1,"2":sd_codes_12_2}
return sd_codes
if n == 14:
# all of these are Type I; one has a unique lowest weight codeword
# (there are 4 total inequiv sd codes of n = 14, by Table 9.10 [HP])
# [14,0]:
genmat = I2(n).augment(I2(n))
# G = PermutationGroup( ["(7,14)", "(6,7)(13,14)", "(5,6)(12,13)", "(4,5)(11,12)",\
# "(3,4)(10,11)", "(2,3)(9,10)", "(1,2)(8,9)"] )
spectrum = [1, 0, 7, 0, 21, 0, 35, 0, 35, 0, 21, 0, 7, 0, 1]
sd_codes_14_0 = {"order autgp":645120,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":"No Type II of this length. Huge aut gp."}
# [14,1]:
genmat = I2(n).augment(block_diagonal_matrix([matA(n)[4],matId(n)[4]]))
# G = PermutationGroup( ["(7,14)", "(6,7)(13,14)", "(5,6)(12,13)", "(4,8)(9,10)",\
# "(4,9)(8,10)", "(3,4)(10,11)", "(2,3)(9,10)", "(1,2)(8,9)"] )
spectrum = [1, 0, 3, 0, 17, 0, 43, 0, 43, 0, 17, 0, 3, 0, 1]
sd_codes_14_1 = {"order autgp":64512,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":"Automorphism group has order 64512."}
# [14,2]:
genmat = I2(n).augment(block_diagonal_matrix([matA(n)[6],matId(n)[6]]))
# G = PermutationGroup( ["(7,14)", "(5,6)(12,13)", "(5,12)(6,13)", "(4,5)(11,12)",\
# "(3,4)(10,11)", "(2,3)(9,10)", "(1,2)(8,9)"] )
spectrum = [1, 0, 1, 0, 15, 0, 47, 0, 47, 0, 15, 0, 1, 0, 1]
sd_codes_14_2 = {"order autgp":46080,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":"Unique codeword of weight 2."}
# [14,3]:
genmat = I2(n).augment(And7)
# G = PermutationGroup( ["(7,11)(12,13)", "(7,12)(11,13)", "(6,9)(10,14)",\
# "(6,10)(9,14)", "(5,6)(8,9)", "(4,5)(9,10), (2,3)(11,12)", "(2,7)(3,13)",\
# "(1,2)(12,13)", "(1,4)(2,5)(3,8)(6,7)(9,13)(10,12)(11,14)"])
spectrum = [1, 0, 0, 0, 14, 0, 49, 0, 49, 0, 14, 0, 0, 0, 1]
sd_codes_14_3 = {"order autgp":56448,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":"Largest minimum distance of these."}
sd_codes["14"] = {"0":sd_codes_14_0,"1":sd_codes_14_1,"2":sd_codes_14_2,\
"3":sd_codes_14_3}
return sd_codes
if n == 16:
# 4 of these are Type I, 2 are Type II. The 2 Type II codes
# are formally equivalent but with different automorphism groups
# [16,0]:
genmat = I2(n).augment(I2(n))
# G = PermutationGroup( [ "(8,16)", "(7,8)(15,16)", "(6,7)(14,15)", "(5,6)(13,14)",
# "(4,5)(12,13)", "(3,4)(11,12)", "(2,3)(10,11)", "(1,2)(9,10)"] )
spectrum = [1, 0, 8, 0, 28, 0, 56, 0, 70, 0, 56, 0, 28, 0, 8, 0, 1]
sd_codes_16_0 = {"order autgp":10321920,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":"Huge aut gp."}
# [16,1]:
genmat = I2(n).augment(block_diagonal_matrix([matA(n)[4],matId(n)[4]]))
# G = PermutationGroup( [ "(8,16)", "(7,8)(15,16)", "(6,7)(14,15)", "(5,6)(13,14)",\
# "(4,9)(10,11)", "(4,10)(9,11)", "(3,4)(11,12)", "(2,3)(10,11)", "(1,2)(9,10)"] )
spectrum = [1, 0, 4, 0, 20, 0, 60, 0, 86, 0, 60, 0, 20, 0, 4, 0, 1]
sd_codes_16_1 = {"order autgp":516096,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":""}
# [16,2]:
genmat = I2(n).augment(block_diagonal_matrix([matA(n)[4],matA(n)[4]]))
# G = PermutationGroup( [ "(8,13)(14,15)", "(8,14)(13,15)", "(7,8)(15,16)", "(6,7)(14,15)",\
# "(5,6)(13,14)", "(4,9)(10,11)", "(4,10)(9,11)", "(3,4)(11,12)", "(2,3)(10,11)",\
# "(1,2)(9,10)","(1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)"] )
spectrum = [1, 0, 0, 0, 28, 0, 0, 0, 198, 0, 0, 0, 28, 0, 0, 0, 1]
sd_codes_16_2 = {"order autgp":3612672,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"II","Comment":"Same spectrum as the other Type II code."}
# [16,3]:
genmat = I2(n).augment(block_diagonal_matrix([matA(n)[6],matId(n)[6]]))
# G = PermutationGroup( [ "(8,16)", "(7,8)(15,16)", "(5,6)(13,14)", "(5,13)(6,14)",\
# "(4,5)(12,13)", "(3,4)(11,12)", "(2,3)(10,11)", "(1,2)(9,10)"] )
spectrum = [1, 0, 2, 0, 16, 0, 62, 0, 94, 0, 62, 0, 16, 0, 2, 0, 1]
sd_codes_16_3 = {"order autgp":184320,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":""}
# [16,4]:
genmat = I2(n).augment(matA(n)[8])
# an equivalent form: See also [20,8] using A[10]
# [(1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1),
# (0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1),
# (0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0),
# (0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0),
# (0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0),
# (0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0),
# (0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0),
# (0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1)]
# G = PermutationGroup( [ "(7,8)(15,16)", "(7,15)(8,16)", "(6,7)(14,15)",\
# "(5,6)(13,14)","(4,5)(12,13)","(3,4)(11,12)", "(2,3)(10,11)", "(1,2)(9,10)"] )
spectrum = [1, 0, 0, 0, 28, 0, 0, 0, 198, 0, 0, 0, 28, 0, 0, 0, 1]
sd_codes_16_4 = {"order autgp":5160960,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"II","Comment":"Same spectrum as the other Type II code. Large aut gp."}
# [16,5]:
genmat = I2(n).augment(block_diagonal_matrix([And7,matId(n)[7]]))
# G = PermutationGroup( [ "(8,16)", "(7,12)(13,14)", "(7,13)(12,14)",\
# "(6,10)(11,15)", "(6,11)(10,15)", "(5,6)(9,10)", "(4,5)(10,11)",\
# "(2,3)(12,13)", "(2,7)(3,14)", "(1,2)(13,14)",\
# "(1,4)(2,5)(3,9)(6,7)(10,14)(11,13)(12,15)" ] )
spectrum = [1, 0, 1, 0, 14, 0, 63, 0, 98, 0, 63, 0, 14, 0, 1, 0, 1]
sd_codes_16_5 = {"order autgp":112896,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":"'Exceptional' construction."}
# [16,6]:
J8 = MatrixSpace(ZZ,8,8)(64*[1])
genmat = I2(n).augment(I2(n)+MS2(n)((H8+J8)/2))
# G = PermutationGroup( [ "(7,9)(10,16)", "(7,10)(9,16)", "(6,7)(10,11)",\
# "(4,6)(11,13)", "(3,5)(12,14)", "(3,12)(5,14)", "(2,3)(14,15)",\
# "(1,2)(8,15)", "(1,4)(2,6)(3,7)(5,16)(8,13)(9,12)(10,14)(11,15)" ] )
spectrum = [1, 0, 0, 0, 12, 0, 64, 0, 102, 0, 64, 0, 12, 0, 0, 0, 1]
sd_codes_16_6 = {"order autgp":73728,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":"'Exceptional' construction. Min dist 4."}
sd_codes["16"] = {"0":sd_codes_16_0,"1":sd_codes_16_1,"2":sd_codes_16_2,\
"3":sd_codes_16_3,"4":sd_codes_16_4,"5":sd_codes_16_5,"6":sd_codes_16_6}
return sd_codes
if n == 18:
# all of these are Type I, all are "extensions" of the n=16 codes
# [18,3] and [18,4] each has a unique lowest weight codeword. Also, they
# are formally equivalent but with different automorphism groups
# [18,0]:
genmat = I2(n).augment(I2(n))
# G = PermutationGroup( [ "(9,18)", "(8,9)(17,18)", "(7,8)(16,17)", "(6,7)(15,16)",\
# "(5,6)(14,15)", "(4,5)(13,14)", "(3,4)(12,13)", "(2,3)(11,12)", "(1,2)(10,11)" ] )
spectrum = [1, 0, 9, 0, 36, 0, 84, 0, 126, 0, 126, 0, 84, 0, 36, 0, 9, 0, 1]
sd_codes_18_0 = {"order autgp":185794560,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment": "Huge aut gp. S_9x(ZZ/2ZZ)^9?"}
# [18,1]:
genmat = I2(n).augment(block_diagonal_matrix([matA(n)[4],matId(n)[4]]))
# G = PermutationGroup( [ "(9,18)", "(8,9)(17,18)", "(7,8)(16,17)", "(6,7)(15,16)",\
# "(5,6)(14,15)", "(4,10)(11,12)", "(4,11)(10,12)", "(3,4)(12,13)",\
# "(2,3)(11,12)", "(1,2)(10,11)" ] )
spectrum = [1, 0, 5, 0, 24, 0, 80, 0, 146, 0, 146, 0, 80, 0, 24, 0, 5, 0, 1]
sd_codes_18_1 = {"order autgp":5160960,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment": "Large aut gp."}
# [18,2]:
genmat = I2(n).augment(block_diagonal_matrix([matA(n)[6],matId(n)[6]]))
# G = PermutationGroup( [ "(9,18)", "(8,9)(17,18)", "(7,8)(16,17)", "(5,6)(14,15)",\
# "(5,14)(6,15)","(4,5)(13,14)", "(3,4)(12,13)", "(2,3)(11,12)", "(1,2)(10,11)"] )
spectrum = [1, 0, 3, 0, 18, 0, 78, 0, 156, 0, 156, 0, 78, 0, 18, 0, 3, 0, 1]
sd_codes_18_2 = {"order autgp":1105920,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment": ""}
# [18,3]:
genmat = I2(n).augment(block_diagonal_matrix([matA(n)[4],matA(n)[4],matId(n)[8]]))
# G = PermutationGroup( [ "(9,18)", "(8,14)(15,16)", "(8,15)(14,16)", "(7,8)(16,17)",\
# "(6,7)(15,16)","(5,6)(14,15)", "(4,10)(11,12)", "(4,11)(10,12)",\
# "(3,4)(12,13)", "(2,3)(11,12)","(1,2)(10,11)",\
# "(1,5)(2,6)(3,7)(4,8)(10,14)(11,15)(12,16)(13,17)" ] )
spectrum = [1, 0, 1, 0, 28, 0, 28, 0, 198, 0, 198, 0, 28, 0, 28, 0, 1, 0, 1]
sd_codes_18_3 = {"order autgp":7225344,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment": "Large aut gp. Unique codeword of smallest non-zero wt.\
Same spectrum as '[18,4]' sd code."}
# [18,4]:
genmat = I2(n).augment(block_diagonal_matrix([matA(n)[8],matId(n)[8]]))
# G = PermutationGroup( [ "(9,18)", "(7,8)(16,17)", "(7,16)(8,17)", "(6,7)(15,16)", \
# "(5,6)(14,15)", "(4,5)(13,14)", "(3,4)(12,13)", "(2,3)(11,12)", "(1,2)(10,11)" ] )
spectrum = [1, 0, 1, 0, 28, 0, 28, 0, 198, 0, 198, 0, 28, 0, 28, 0, 1, 0, 1]
sd_codes_18_4 = {"order autgp":10321920,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment": "Huge aut gp. Unique codeword of smallest non-zero wt.\
Same spectrum as '[18,3]' sd code."}
# [18,5]:
C = self_dual_codes_binary(n-2)["%s"%(n-2)]["5"]["code"]
A0 = C.redundancy_matrix()
genmat = I2(n).augment(block_diagonal_matrix([A0,matId(n)[8]]))
# G = PermutationGroup( [ "(5,10)(6,11)", "(5,11)(6,10)", "(5,11,12)(6,7,10)",\
# "(5,11,10,7,12,6,13)", "(2,15)(3,16)(5,11,10,7,12,6,13)",\
# "(2,16)(3,15)(5,11,10,7,12,6,13)", "(2,16,14)(3,15,4)(5,11,10,7,12,6,13)",\
# "(1,2,16,15,4,3,14)(5,11,10,7,12,6,13)", "(1,5,14,6,16,11,15,7,3,10,4,12,2,13)",\
# "(2,16,14)(3,15,4)(5,11,10,7,12,6,13)(9,18)",\
# "(2,16,14)(3,15,4)(5,11,10,7,12,6,13)(8,9,17,18)" ] )
spectrum = [1, 0, 2, 0, 15, 0, 77, 0, 161, 0, 161, 0, 77, 0, 15, 0, 2, 0, 1]
sd_codes_18_5 = {"order autgp":451584,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment": "'Exceptional' construction."}
# [18,6]:
C = self_dual_codes_binary(n-2)["%s"%(n-2)]["6"]["code"]
A0 = C.redundancy_matrix()
genmat = I2(n).augment(block_diagonal_matrix([A0,matId(n)[8]]))
G = PermutationGroup( [ "(9,18)", "(7,10)(11,17)", "(7,11)(10,17)", "(6,7)(11,12)",\
"(4,6)(12,14)", "(3,5)(13,15)", "(3,13)(5,15)", "(2,3)(15,16)", "(1,2)(8,16)",\
"(1,4)(2,6)(3,7)(5,17)(8,14)(10,13)(11,15)(12,16)" ] )
spectrum = [1, 0, 1, 0, 12, 0, 76, 0, 166, 0, 166, 0, 76, 0, 12, 0, 1, 0, 1]
sd_codes_18_6 = {"order autgp":147456,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment": "'Exceptional'. Unique codeword of smallest non-zero wt."}
# [18,7] (equiv to H18 in [P])
genmat = MS(n)([[1,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0],\
[0,1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,1],\
[0,0,1,0,0,0,0,0,0,1,1,1,0,0,1,0,0,1],\
[0,0,0,1,0,0,0,0,0,1,1,1,1,0,0,0,0,1],\
[0,0,0,0,1,0,0,0,0,1,1,0,0,1,0,1,1,0],\
[0,0,0,0,0,1,0,0,0,1,0,1,0,1,0,1,1,0],\
[0,0,0,0,0,0,1,0,0,0,1,1,0,0,0,0,1,0],\
[0,0,0,0,0,0,0,1,0,1,0,0,0,1,0,0,0,1],\
[0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,1]])
# G = PermutationGroup( [ "(9,10)(16,18)", "(9,16)(10,18)", "(8,9)(14,16)",\
# "(7,11)(12,17)", "(7,12)(11,17)", "(5,6)(11,12)", "(5,7)(6,17)",\
# "(4,13)(5,8)(6,14)(7,9)(10,12)(11,18)(16,17)", "(3,4)(13,15)",\
# "(1,2)(5,8)(6,14)(7,9)(10,12)(11,18)(16,17)", "(1,3)(2,15)",\
# "(1,5)(2,6)(3,7)(4,11)(10,18)(12,13)(15,17)" ] )
spectrum = [1, 0, 0, 0, 9, 0, 75, 0, 171, 0, 171, 0, 75, 0, 9, 0, 0, 0, 1]
sd_codes_18_7 = {"order autgp":82944,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment": "'Exceptional' construction. Min dist 4."}
# [18, 8] (equiv to I18 in [P])
I18 = MS(n)([[1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],\
[0,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0],\
[0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0],\
[0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0],\
[1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0],\
[0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0],\
[0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0],\
[0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,1],\
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]])
genmat = MS(n)([[1,0,0,0,0,0,0,0,0, 1, 1, 1, 1, 1, 0, 0, 0, 0],\
[0,1,0,0,0,0,0,0,0, 1, 0, 1, 1, 1, 0, 1, 1, 1],\
[0,0,1,0,0,0,0,0,0, 0, 1, 1, 0, 0, 0, 1, 1, 1],\
[0,0,0,1,0,0,0,0,0, 0, 1, 0, 0, 1, 0, 1, 1, 1],\
[0,0,0,0,1,0,0,0,0, 0, 1, 0, 1, 0, 0, 1, 1, 1],\
[0,0,0,0,0,1,0,0,0, 1, 1, 0, 0, 0, 0, 1, 1, 1],\
[0,0,0,0,0,0,1,0,0, 0, 0, 0, 0, 0, 1, 0, 1, 1],\
[0,0,0,0,0,0,0,1,0, 0, 0, 0, 0, 0, 1, 1, 0, 1],\
[0,0,0,0,0,0,0,0,1, 0, 0, 0, 0, 0, 1, 1, 1, 0]])
G = PermutationGroup( [ "(9,15)(16,17)", "(9,16)(15,17)", "(8,9)(17,18)",\
"(7,8)(16,17)", "(5,6)(10,13)", "(5,10)(6,13)", "(4,5)(13,14)",\
"(3,4)(12,14)", "(1,2)(6,10)", "(1,3)(2,12)" ] )
spectrum = [1, 0, 0, 0, 17, 0, 51, 0, 187, 0, 187, 0, 51, 0, 17, 0, 0, 0, 1]
sd_codes_18_8 = {"order autgp":322560,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment": "'Exceptional' construction. Min dist 4."}
sd_codes["18"] = {"0":sd_codes_18_0,"1":sd_codes_18_1,"2":sd_codes_18_2,\
"3":sd_codes_18_3,"4":sd_codes_18_4,"5":sd_codes_18_5,\
"6":sd_codes_18_6,"7":sd_codes_18_7,"8":sd_codes_18_8}
return sd_codes
if n == 20:
# all of these of these are Type I; 2 of these codes
# are formally equivalent but with different automorphism groups;
# one of these has a unique codeword of lowest weight
A10 = MatrixSpace(F,10,10)([[1, 1, 1, 1, 1, 1, 1, 1, 1, 0],\
[1, 1, 1, 0, 1, 0, 1, 0, 1, 1],\
[1, 0, 0, 1, 0, 1, 0, 1, 0, 1],\
[0, 0, 0, 1, 1, 1, 0, 1, 0, 1],\
[0, 0, 1, 1, 0, 1, 0, 1, 0, 1],\
[0, 0, 0, 1, 0, 1, 1, 1, 0, 1],\
[0, 1, 0, 1, 0, 1, 0, 1, 0, 1],\
[0, 0, 0, 1, 0, 0, 0, 0, 1, 1],\
[0, 0, 0, 0, 0, 1, 0, 0, 1, 1],\
[0, 0, 0, 0, 0, 0, 0, 1, 1, 1]])
# [20,0]:
genmat = I2(n).augment(I2(n))
# G = PermutationGroup( ["(10,20)", "(9,10)(19,20)", "(8,9)(18,19)", "(7,8)(17,18)", "(6,7)(16,17)",\
# "(5,6)(15,16)", "(4,5)(14,15)", "(3,4)(13,14)", "(2,3)(12,13)", "(1,2)(11,12)"] )
spectrum = [1, 0, 10, 0, 45, 0, 120, 0, 210, 0, 252, 0, 210, 0, 120, 0, 45, 0, 10, 0, 1]
sd_codes_20_0 = {"order autgp":3715891200,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment": "Huge aut gp"}
# [20,1]:
genmat = I2(n).augment(block_diagonal_matrix([matA(n)[4],matId(n)[4]]))
# G = PermutationGroup( [ "(10,20)", "(9,10)(19,20)", "(8,9)(18,19)", "(7,8)(17,18)", "(6,7)(16,17)",\
# "(5,6)(15,16)", "(4,11)(12,13)", "(4,12)(11,13)", "(3,4)(13,14)",\
# "(2,3)(12,13)", "(1,2)(11,12)"] )
spectrum = [1, 0, 6, 0, 29, 0, 104, 0, 226, 0, 292, 0, 226, 0, 104, 0, 29, 0, 6, 0, 1]
sd_codes_20_1 = {"order autgp":61931520,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":""}
# [20,2]:
genmat = I2(n).augment(block_diagonal_matrix([matA(n)[6],matId(n)[6]]))
# G = PermutationGroup( [ "(10,20)", "(9,10)(19,20)", "(8,9)(18,19)", "(7,8)(17,18)",\
# "(5,6)(15,16)", "(5,15)(6,16)", "(4,5)(14,15)", "(3,4)(13,14)",\
# "(2,3)(12,13)", "(1,2)(11,12)"] )
spectrum = [1, 0, 4, 0, 21, 0, 96, 0, 234, 0, 312, 0, 234, 0, 96, 0, 21, 0, 4, 0, 1]
sd_codes_20_2 = {"order autgp":8847360,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":""}
# [20,3]:
genmat = I2(n).augment(block_diagonal_matrix([matA(n)[6],matA(n)[4]]))
# G = PermutationGroup( [ "(5,6)(15,16)", "(5,15)(6,16)", "(4,5)(14,15)", "(3,4)(13,14)",\
# "(2,3)(12,13)", "(1,2)(11,12)", "(8,17)(9,10)", "(8,10)(9,17)", "(8,10,20)(9,19,17)",\
# "(8,19,20,9,17,10,18)", "(7,8,19,20,9,18)(10,17)"] )
spectrum =[1, 0, 0, 0, 29, 0, 32, 0, 226, 0, 448, 0, 226, 0, 32, 0, 29, 0, 0, 0, 1]
sd_codes_20_3 = {"order autgp":30965760,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":"Min dist 4."}
# [20,4]:
genmat = I2(n).augment(block_diagonal_matrix([matA(n)[4],matA(n)[4],matId(n)[8]]))
# G = PermutationGroup( [ "(5,15)(6,16)", "(5,16)(6,15)", "(5,16,7)(6,17,15)", "(5,15,8)(6,17,7)",\
# "(5,17,18)(6,15,8), (3,14)(4,13)(5,17,18)(6,15,8)", "(3,13)(4,14)(5,17,18)(6,15,8)",\
# "(2,3,14)(4,13,11)(5,17,18)(6,15,8)"," (2,3,12)(4,11,14)(5,17,18)(6,15,8)",\
# "(1,2,3,11,14,4,12)(5,17,18)(6,15,8)", "(1,5,13,17,14,8,2,7,3,16,12,6,11,18)(4,15)",\
# "(2,3,12)(4,11,14)(5,17,18)(6,15,8)(10,20)",\
# "(2,3,12)(4,11,14)(5,17,18)(6,15,8)(9,10,19,20)"] )
spectrum =[1, 0, 2, 0, 29, 0, 56, 0, 226, 0, 396, 0, 226, 0, 56, 0, 29, 0, 2, 0, 1]
sd_codes_20_4 = {"order autgp":28901376,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":""}
# [20,5]:
genmat = I2(n).augment(block_diagonal_matrix([And7,matId(n)[7]]))
# G = PermutationGroup( [ "(10,20)", "(9,10)(19,20)", "(8,9)(18,19)",\
# "(7,11)(12,14)", "(7,12)(11,14)", "(6,7)(12,13)", "(5,6)(11,12)",\
# "(4,15)(16,17)", "(4,16)(15,17)", "(2,3)(16,17)", "(2,4)(3,15)",\
# "(1,2)(15,16)", "(1,5)(2,6)(3,13)(4,7)(11,16)(12,15)(14,17)" ] ) # order 2709504
spectrum = [1, 0, 3, 0, 17, 0, 92, 0, 238, 0, 322, 0, 238, 0, 92, 0, 17, 0, 3, 0, 1]
sd_codes_20_5 = {"order autgp":2709504,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment": "'Exceptional' construction."}
# [20,6]:
genmat = I2(n).augment(block_diagonal_matrix([matA(n)[8],matId(n)[8]]))
# G = PermutationGroup( [ "(7,8)(17,18)", "(7,17)(8,18)", "(6,7)(16,17)", "(5,6)(15,16)",\
# "(4,5)(14,15)", "(3,4)(13,14)", "(2,3)(12,13)", "(1,2)(11,12)",\
# "(10,20)", "(9,10,19,20)"] )
spectrum = [1, 0, 2, 0, 29, 0, 56, 0, 226, 0, 396, 0, 226, 0, 56, 0, 29, 0, 2, 0, 1]
sd_codes_20_6 = {"order autgp":41287680,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":""}
# [20,7]:
A0 = self_dual_codes_binary(n-4)["16"]["6"]["code"].redundancy_matrix()
genmat = I2(n).augment(block_diagonal_matrix([A0,matId(n)[8]]))
# G = PermutationGroup( [ "(10,20)", "(9,10)(19,20)", "(7,11)(12,18)",\
# "(7,12)(11,18)", "(6,7)(12,13)", "(4,6)(13,15)", "(3,5)(14,16)",\
# "(3,14)(5,16)", "(2,3)(16,17)", "(1,2)(8,17)",\
# "(1,4)(2,6)(3,7)(5,18)(8,15)(11,14)(12,16)(13,17)" ] )
spectrum = [1,0,2,0,13,0,88,0,242,0,332,0,242,0,88,0,13,0,2,0,1]
sd_codes_20_7 = {"order autgp":589824,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":"'Exceptional' construction."}
# [20,8]: (genmat, J20, and genmat2 are all equiv)
genmat = I2(n).augment(matA(n)[10])
J20 = MS(n)([[1,1,1, 1, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0],\
[0,0,0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 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],\
[0,0,0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0],\
[0,0,0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1],\
[1,0,1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0]])
genmat2 = MS(n)([[1,0,0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1],\
[0,1,0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1],\
[0,0,1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0],\
[0,0,0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0],\
[0,0,0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0],\
[0,0,0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0],\
[0,0,0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0],\
[0,0,0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0],\
[0,0,0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0],\
[0,0,0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1]])
# G = PermutationGroup( [ "(9,10)(19,20)", "(9,19)(10,20)", "(8,9)(18,19)", "(7,8)(17,18)",\
# "(6,7)(16,17)", "(5,6)(15,16)", "(4,5)(14,15)", "(3,4)(13,14)",\
# "(2,3)(12,13)", "(1,2)(11,12)"] )
spectrum =[1, 0, 0, 0, 45, 0, 0, 0, 210, 0, 512, 0, 210, 0, 0, 0, 45, 0, 0, 0, 1]
sd_codes_20_8 = {"order autgp":1857945600,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":"Huge aut gp. Min dist 4."}
# [20,9]: (genmat, K20 are equiv)
genmat = I2(n).augment(A10)
K20 = MS(n)([[1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],\
[0,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],\
[0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0],\
[0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0],\
[0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0],\
[0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0],\
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0],\
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1],\
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0],\
[0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0]])
#genmat = K20 # not in standard form
# G = PermutationGroup( [ "(4,13)(5,15)", "(4,15)(5,13)", "(3,4,13)(5,11,15)",
# "(3,4,6,11,15,17)(5,13)", "(3,5,17,4,12)(6,15,7,11,13)",
# "(1,2)(3,5,17,4,7,11,13,6,15,12)", "(1,3,5,17,4,12)(2,11,13,6,15,7)",
# "(3,5,17,4,12)(6,15,7,11,13)(10,18)(19,20)", "(3,5,17,4,12)(6,15,7,11,13)(10,19)(18,20)",
# "(3,5,17,4,12)(6,15,7,11,13)(9,10)(16,18)",
# "(3,5,17,4,12)(6,15,7,11,13)(8,9)(14,16)" ] )
spectrum = [1, 0, 0, 0, 21, 0, 48, 0, 234, 0, 416, 0, 234, 0, 48, 0, 21, 0, 0, 0, 1]
sd_codes_20_9 = {"order autgp":4423680,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment": "Min dist 4."}
# [20,10]
L20 = MS(n)([[1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],\
[0,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],\
[1,0,1,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0],\
[0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0,0],\
[0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0],\
[0,0,0,0,0,0,0,1,0,1,0,1,0,1,0,0,0,0,0,0],\
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0],\
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1],\
[0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0,0,0],\
[0,1,0,1,0,1,0,0,0,0,0,0,0,0,1,0,1,0,1,0]])
genmat = L20 # not in standard form
# G = PermutationGroup( [ "(17,18)(19,20)", "(17,19)(18,20)", "(15,16)(19,20)",
# "(15,17)(16,18)", "(10,11)(12,13)", "(10,12)(11,13)", "(9,10)(13,14)",
# "(8,9)(12,13)", "(3,4)(5,6)", "(3,5)(4,6)", "(2,3)(6,7)", "(1,2)(5,6)",
# "(1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(19,20)" ] ) # order 1354752
spectrum = [1, 0, 0, 0, 17, 0, 56, 0, 238, 0, 400, 0, 238, 0, 56, 0, 17, 0, 0, 0, 1]
sd_codes_20_10 = {"order autgp":1354752,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment": "Min dist 4."}
# [20,11]
S20 = MS(n)([[1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],\
[0,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],\
[0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0],\
[0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0],\
[0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0],\
[0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0],\
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1],\
[1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,1,1,0,0],\
[1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0],\
[1,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,1,0]] )
genmat = S20 # not in standard form
# G = PermutationGroup( [ "(17,18)(19,20)", "(17,19)(18,20)", "(13,14)(15,16)",
# "(13,15)(14,16)", "(11,12)(15,16)", "(11,13)(12,14)", "(9,10)(15,16)",
# "(9,11)(10,12)", "(5,6)(7,8)", "(5,7)(6,8)", "(3,4)(7,8)", "(3,5)(4,6)",
# "(1,2)(7,8)", "(1,3)(2,4)", "(1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)" ] )
# G.order() = 294912
spectrum = [1, 0, 0, 0, 13, 0, 64, 0, 242, 0, 384, 0, 242, 0, 64, 0, 13, 0, 0, 0, 1]
sd_codes_20_11 = {"order autgp":294912,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":"Min dist 4."}
# [20,12]
R20 = MS(n)([[0,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],\
[0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0],\
[0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0],\
[0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0],\
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0],\
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1],\
[0,1,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,1,1,0],\
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0],\
[1,1,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1],\
[1,1,0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0,1,1]])
genmat = R20 # not in standard form
# G = PermutationGroup( [ "(17,18)(19,20)", "(17,19)(18,20)", "(15,16)(19,20)",
# "(15,17)(16,18)", "(11,12)(13,14)", "(11,13)(12,14)", "(9,10)(13,14)",
# "(9,11)(10,12)", "(5,6)(7,8)", "(5,7)(6,8)", "(3,4)(7,8)", "(3,5)(4,6)",
# "(3,9,15)(4,10,16)(5,11,17)(6,12,18)(7,14,19)(8,13,20)",
# "(1,2)(7,8)(9,15)(10,16)(11,17)(12,18)(13,19)(14,20)" ] ) # order 82944
spectrum = [1, 0, 0, 0, 9, 0, 72, 0, 246, 0, 368, 0, 246, 0, 72, 0, 9, 0, 0, 0, 1]
sd_codes_20_12 = {"order autgp":82944,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":"Min dist 4."}
# [20,13]
M20 = MS(n)([[1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],\
[0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0],\
[0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0],\
[0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0],\
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1],\
[0,0,0,0,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0],\
[1,1,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,1,1,0],\
[0,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,1],\
[0,0,1,1,0,1,1,0,0,0,1,1,0,0,0,0,0,0,0,0],\
[0,0,0,0,0,0,1,1,0,1,1,0,1,0,0,1,0,0,0,0]])
genmat = M20 # not in standard form
# G = PermutationGroup( [ "(17,18)(19,20)", "(17,19)(18,20)", "(13,14)(15,16)",
# "(13,15)(14,16)", "(9,10)(11,12)", "(9,11)(10,12)", "(5,6)(7,8)",
# "(5,7)(6,8)", "(5,9)(6,11)(7,12)(8,10)(13,17)(14,19)(15,18)(16,20)",
# "(5,13)(6,15)(7,14)(8,16)(9,17)(10,20)(11,18)(12,19)",
# "(3,4)(6,7)(11,12)(13,17)(14,18)(15,19)(16,20)",
# "(2,3)(7,8)(9,13)(10,14)(11,15)(12,16)(19,20)",
# "(1,2)(6,7)(11,12)(13,17)(14,18)(15,19)(16,20)",
# "(1,5)(2,6)(3,7)(4,8)(9,17)(10,18)(11,19)(12,20)" ] )
spectrum = [1, 0, 0, 0, 5, 0, 80, 0, 250, 0, 352, 0, 250, 0, 80, 0, 5, 0, 0, 0, 1]
sd_codes_20_13 = {"order autgp":122880,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment": "Min dist 4."}
# [20,14]: # aut gp of this computed using a program by Robert Miller
A0 = self_dual_codes_binary(n-2)["18"]["8"]["code"].redundancy_matrix()
genmat = I2(n).augment(block_diagonal_matrix([A0,matId(n)[9]]))
# [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0],
# [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0],
# [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0],
# [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0],
# [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0],
# [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0],
# [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0],
# [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0],
# [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0],
# [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]]
# G = PermutationGroup( [ "(8,19)(16,17)", "(8,16)(17,19)", "(9,18)(16,17)", "(8,9)(18,19)",
# "(7,8)(17,18)", "(4,15)(5,14)", "(4,5)(14,15)", "(4,15)(6,11)", "(5,6)(11,14)",
# "(3,13)(4,15)", "(3,15)(4,13)", "(1,2)(4,15)", "(1,4)(2,15)(3,5)(13,14)", "(10,20)" ] )
spectrum = [1, 0, 1, 0, 17, 0, 68, 0, 238, 0, 374, 0, 238, 0, 68, 0, 17, 0, 1, 0, 1]
sd_codes_20_14 = {"order autgp":645120,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment": "'Exceptional' construction."}
# [20,15]:
A0 = self_dual_codes_binary(n-2)["18"]["7"]["code"].redundancy_matrix()
genmat = I2(n).augment(block_diagonal_matrix([A0,matId(n)[9]]))
# [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0],
# [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0],
# [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0],
# [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0],
# [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0],
# [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0],
# [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0],
# [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0],
# [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0],
# [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]]
# G = PermutationGroup( [ "(10,20)", "(9,11)(17,19)", "(9,17)(11,19)", "(8,9)(15,17)",
# "(7,12)(13,18)", "(7,13)(12,18)", "(5,6)(12,13)", "(5,7)(6,18)",
# "(4,14)(5,8)(6,15)(7,9)(11,13)(12,19)(17,18)", "(3,4)(14,16)",
# "(1,2)(5,8)(6,15)(7,9)(11,13)(12,19)(17,18)", "(1,3)(2,16)",
# "(1,5)(2,6)(3,7)(4,12)(11,19)(13,14)(16,18)" ] ) # order 165888
spectrum = [1, 0, 1, 0, 9, 0, 84, 0, 246, 0, 342, 0, 246, 0, 84, 0, 9, 0, 1, 0, 1]
sd_codes_20_15 = {"order autgp":165888,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":"'Exceptional' construction. Unique lowest wt codeword."}
sd_codes["20"] = {"0":sd_codes_20_0,"1":sd_codes_20_1,"2":sd_codes_20_2,\
"3":sd_codes_20_3,"4":sd_codes_20_4,"5":sd_codes_20_5,\
"6":sd_codes_20_6,"7":sd_codes_20_7,"8":sd_codes_20_8,\
"9":sd_codes_20_9,"10":sd_codes_20_10,"11":sd_codes_20_11,\
"12":sd_codes_20_12,"13":sd_codes_20_13,"14":sd_codes_20_14,
"15":sd_codes_20_15}
return sd_codes
if n == 22:
# all of these of these are Type I; 2 of these codes
# are formally equivalent but with different automorphism groups
# *** Incomplete *** (7 out of 25)
# [22,0]:
genmat = I2(n).augment(I2(n))
# G = PermutationGroup( [ "(11,22)", "(10,11)(21,22)", "(9,10)(20,21)",\
# "(8,9)(19,20)", "(7,8)(18,19)", "(6,7)(17,18)", "(5,6)(16,17)",\
# "(4,5)(15,16)", "(3,4)(14,15)", "(2,3)(13,14)", "(1,2)(12,13)" ] ) # S_11x(ZZ/2ZZ)^11??
spectrum = [1, 0, 11, 0, 55, 0, 165, 0, 330, 0, 462, 0, 462, 0, 330, 0, 165, 0, 55, 0, 11, 0, 1]
sd_codes_22_0 = {"order autgp":81749606400,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":"Huge aut gp."}
# [22,1]:
genmat = I2(n).augment(block_diagonal_matrix([matA(n)[4],matId(n)[4]]))
# G = PermutationGroup( [ "(11,22)", "(10,11)(21,22)", "(9,10)(20,21)",\
# "(8,9)(19,20)", "(7,8)(18,19)", "(6,7)(17,18)", "(5,6)(16,17)",\
# "(4,12)(13,14)", "(4,13)(12,14)", "(3,4)(14,15)", "(2,3)(13,14)", "(1,2)(12,13)" ] )
spectrum = [1, 0, 7, 0, 35, 0, 133, 0, 330, 0, 518, 0, 518, 0, 330, 0, 133, 0, 35, 0, 7, 0, 1]
sd_codes_22_1 = {"order autgp":867041280,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":""}
# [22,2]:
genmat = I2(n).augment(block_diagonal_matrix([matA(n)[6],matId(n)[6]]))
# G = PermutationGroup( [ "(11,22)", "(10,11)(21,22)", "(9,10)(20,21)",\
# "(8,9)(19,20)", "(7,8)(18,19)", "(5,6)(16,17)", "(5,16)(6,17)",\
# "(4,5)(15,16)", "(3,4)(14,15)", "(2,3)(13,14)", "(1,2)(12,13)" ] )
spectrum = [1, 0, 5, 0, 25, 0, 117, 0, 330, 0, 546, 0, 546, 0, 330, 0, 117, 0, 25, 0, 5, 0, 1]
sd_codes_22_2 = {"order autgp":88473600,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":""}
# [22,3]:
genmat = I2(n).augment(block_diagonal_matrix([matA(n)[8],matId(n)[8]]))
# G = PermutationGroup( [ "(11,22)", "(10,11)(21,22)", "(9,10)(20,21)",\
# "(7,8)(18,19)", "(7,18)(8,19)", "(6,7)(17,18)", "(5,6)(16,17)",\
# "(4,5)(15,16)", "(3,4)(14,15)", "(2,3)(13,14)", "(1,2)(12,13)" ] )
spectrum = [1, 0, 3, 0, 31, 0, 85, 0, 282, 0, 622, 0, 622, 0, 282, 0, 85, 0, 31, 0, 3, 0, 1]
sd_codes_22_3 = {"order autgp":247726080,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":"Same spectrum as the '[20,5]' code."}
# [22,4]:
genmat = I2(n).augment(block_diagonal_matrix([matA(n)[10],matId(n)[10]]))
# G = PermutationGroup( [ "(11,22)", "(9,10)(20,21)", "(9,20)(10,21)",\
# "(8,9)(19,20)", "(7,8)(18,19)", "(6,7)(17,18)", "(5,6)(16,17)",\
# "(4,5)(15,16)", "(3,4)(14,15)", "(2,3)(13,14)", "(1,2)(12,13)" ] )
spectrum = [1, 0, 1, 0, 45, 0, 45, 0, 210, 0, 722, 0, 722, 0, 210, 0, 45, 0, 45, 0, 1, 0, 1]
sd_codes_22_4 = {"order autgp":3715891200,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":"Unique lowest weight codeword."}
# [22,5]:
genmat = I2(n).augment(block_diagonal_matrix([matA(n)[4],matA(n)[4],matId(n)[8]]))
# G = PermutationGroup( [ "(11,22)", "(10,11)(21,22)", "(9,10)(20,21)",\
# "(8,16)(17,18)", "(8,17)(16,18)", "(7,8)(18,19)", "(6,7)(17,18)",\
# "(5,6)(16,17)", "(4,12)(13,14)", "(4,13)(12,14)", "(3,4)(14,15)",\
# "(2,3)(13,14)", "(1,2)(12,13)", "(1,5)(2,6)(3,7)(4,8)(12,16)(13,17)(14,18)(15,19)" ] )
spectrum = [1, 0, 3, 0, 31, 0, 85, 0, 282, 0, 622, 0, 622, 0, 282, 0, 85, 0, 31, 0, 3, 0, 1]
sd_codes_22_5 = {"order autgp":173408256,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":"Same spectrum as the '[20,3]' code."}
# [22,6]:
genmat = I2(n).augment(block_diagonal_matrix([matA(n)[6],matA(n)[4],matId(n)[10]]))
# G = PermutationGroup( [ "(11,22)", "(10,18)(19,20)", "(10,19)(18,20)",\
# "(9,10)(20,21)", "(8,9)(19,20)", "(7,8)(18,19)", "(5,6)(16,17)",\
# "(5,16)(6,17)", "(4,5)(15,16)", "(3,4)(14,15)", "(2,3)(13,14)", "(1,2)(12,13)" ] )
spectrum = [1, 0, 1, 0, 29, 0, 61, 0, 258, 0, 674, 0, 674, 0, 258, 0, 61, 0, 29, 0, 1, 0, 1]
sd_codes_22_6 = {"order autgp":61931520,"code":LinearCode(genmat),"spectrum":spectrum,\
"Type":"I","Comment":"Unique lowest weight codeword."}
sd_codes["22"] = {"0":sd_codes_22_0,"1":sd_codes_22_1,"2":sd_codes_22_2,\
"3":sd_codes_22_3,"4":sd_codes_22_4,"5":sd_codes_22_5,\
"6":sd_codes_22_6}
return sd_codes
