def test_cocode():
print("")
for i in range(200):
# Test power map, inveriosn, sign, theta, and conversion to GCode
n1 = randint(0, 0x1fff)
p1 = PLoop(n1)
ccvector = randint(0, 0xffffff)
coc = mat24.vect_to_cocode(ccvector)
cclist = [i for i in range(24) if (ccvector >> i) & 1]
cc1 = Cocode(cclist)
cc2 = Cocode(coc)
if i < 1:
print("\nTesting", GcVector(ccvector), ", cocode =", cc1)
assert cc1 == cc2
u = Parity(mat24.scalar_prod(p1.value, cc1.value))
assert p1 & cc1 == u == cc1 & p1 == u * 1 == u + 0
par = Parity(randint(0, 1))
assert cc1 + par == par + cc1 == cc1.value // 0x800 + par
assert cc1 % 2 == Parity(cc1)
assert len(cc1) == mat24.cocode_weight(cc1.value)
if len(cc1) < 4:
syndrome = mat24.cocode_syndrome(cc1.value)
assert cc1.syndrome().value == syndrome
syn_from_list = sum(1 << i
for i in GcVector(ccvector).syndrome_list())
assert syn_from_list == syndrome
i = randint(0, 23)
assert cc1.syndrome(i).value == mat24.cocode_syndrome(cc1.value, i)
syndrome_list = cc1.syndrome(i).bit_list
assert len(cc1) == len(syndrome_list)
assert syndrome_list == mat24.cocode_to_bit_list(cc1.value, i)
def ref_gen_leech2_reduce_n(v, verbose=0):
vtype = gen_leech2_subtype(v)
subtype = vtype & 0xf
out = []
if subtype & 1:
coc = (v ^ mat24.ploop_theta(v >> 12)) & 0xfff
syn = mat24.cocode_syndrome(coc)
src = mat24.vect_to_list(syn, mat24.bw24(syn))
assert len(src) in [1, 3]
lst = [1, 2, 3] if subtype == 3 else [0]
apply_perm(v, src, lst, len(src), out)
v = gen_leech2_op_atom(v, out[0])
out.append(0xC0000000 + ((v >> 12) & 0x7ff))
v = gen_leech2_op_atom(v, out[1])
out.append(0xB0000000 + ((v >> 13) & 0x800))
elif subtype == 6:
gv = (v >> 12) & 0xfff
vect = mat24.gcode_to_vect(gv)
src = mat24.vect_to_list(vect, mat24.bw24(vect))
assert len(src) == 12
dest = STD_DODECAD
if (vtype == 0x36):
coc = (v ^ mat24.ploop_theta(v >> 12)) & 0xfff
w = mat24.bw24(mat24.cocode_as_subdodecad(coc, gv))
if w & 2:
dest = CPL_DODECAD
pi = mat24.perm_from_dodecads(dest, src)
out.append(0xA0000000 + mat24.perm_to_m24num(pi))
op_y_x(v, TABLE_DODECAD, out)
elif subtype in [2, 4]:
if vtype == 0x34:
find_octad_permutation_odd(v, out)
else:
find_octad_permutation(v, out)
op_y_x(v, TABLE_OCTAD, out)
elif subtype in [0, 8]:
if ((v & 0x7ff) == 0):
out.append(0xA0000000)
out.append(0xC0000000)
x = 0x800 if v & 0x1800000 == 0x1800000 else 0
out.append(0x90000000 + x)
else:
syn = mat24.cocode_syndrome(v & 0x7ff, 0)
src = mat24.vect_to_list(syn, mat24.bw24(syn))
j = mat24.bw24(syn) & 2
lst, y0 = ([2, 3], 0x200) if j else ([0, 1, 2, 3], 0x400)
apply_perm(v, src, lst, len(lst), out)
v = gen_leech2_op_atom(v, out[0])
y = y0 if v & 0x800000 else 0
out.append(0xC0000000 + y)
v = gen_leech2_op_atom(v, out[1])
x = y0 if v & 0x1000000 else 0
out.append(0xB0000000 + x)
else:
raise ValueError("Bad subtype " + hex(vtype))
assert len(out) == 3
return vtype, np.array(out, dtype=np.uint32)
def suboctad_type(octad, w, coc):
"""Return suboctad type.
Let ``octad`` be an octad, i.e. a bit vector of length 8. Let
w = 1 (mod 2) if ``octad`` denotes an octad and ``w = 0``
if ``octad`` denotes a complemented octad. Let ``coc`` be
an even cocode vector in cocode representation.
The function returns 0 in bit 1 of the return value if the
cocode word ``coc`` can be written as a subset of the octad,
and 1 in bit 1 otherwise.
the function returns ``1 + w + bit_weight(coc)/2`` in bit 0
of the return value.
Then Leech lattice vector ``x_octad + x_coc`` is of subtype
0x22 if the return value is zero. Otherwise it is of
subtype 0x44 (or 0x46) if bit 1 of the return value is
(or 1).
"""
lsb = mat24.lsbit24(octad) # ls bit of octad
syn = mat24.cocode_syndrome(coc, lsb) # cocode syndrome
wsub = octad & syn == syn # wsub = 1 if coc is suboctad
cw = mat24.cocode_weight(coc) >> 1 # cw = cocode_weight(v) / 2
return 2 * (1 - wsub) + ((w ^ cw ^ 1) & 1)
def syndrome(self, i=None):
"""Return syndrome of cocode element as a bit vector.
:param i: ``None`` (default) or an integer ``0 <= i < 24``
used to select a syndrome.
:return: Syndrome of a Golay cocode element as a bit
vector of type |GcVector|.
:raise:
* ValueError if argument ``i`` is ``None`` and the
syndrome is not unique.
Any Golay cocode element has either a unique shortest
representative of bit weight ``<= 3`` or precisely six
shortest representatives of bit weight ``4`` which form
a partition of the underlying set of the code. Such a
partition is called a *tetrad*.
In coding theory, a shortest representative of a cocode
element is called a *syndrome*.
If the syndrome is unique, the function returns that
syndrome. Otherwise it returns the syndrome containing
bit at position ``i``.
"""
if i is None: i = 24
return GcVector(mat24.cocode_syndrome(self.value, i))
def find_octad_permutation_odd(v, result, verbose=0):
""" Find a suitable permutation for an octad.
Similar to function ``find_octad_permutation`` in module
``mmgroup.dev.generators.gen_leech_reduce_n``.
Here ``v, o, c`` are as in that function; but the scalar
product of ``o`` and ``c`` must be 1. Apart from that
operation is as in function ``find_octad_permutation``.
We compute a permutation that maps octad ``o`` to the standard
octad (0,1,2,3,4,5,6,7). If the cocode part ``c`` of ``v`` is
not a suboctad of octad ``o`` then we map (one shortest
representative of) ``c`` into the set (0,1,2,3,...7,8).
"""
coc = (v ^ mat24.ploop_theta(v >> 12)) & 0xfff
w = mat24.gcode_weight(v >> 12)
vect = mat24.gcode_to_vect((v ^ ((w & 4) << 21)) >> 12)
src = mat24.vect_to_list(vect, 5)
if mat24.cocode_weight(coc) == 4:
sextet = mat24.cocode_to_sextet(coc)
for i in range(0, 24, 4):
syn = (1 << sextet[i]) | (1 << sextet[i + 1])
syn |= (1 << sextet[i + 2]) | (1 << sextet[i + 3])
special = syn & vect
if special & (special - 1):
break
else:
syn = mat24.cocode_syndrome(coc, 24)
src.append(mat24.lsbit24(syn & ~vect))
return apply_perm(v, src, OCTAD_PLUS, 6, result, verbose)
def xi_reduce_odd_type4(v, verbose=0):
r"""Compute power of :math:`\xi` that reduces a vector ``v``
Let ``v`` be a vector in the Leech lattice mod 2 in Leech
lattice encoding. We assume that ``v`` is of subtype 0x43.
We compute an exponent ``e`` such that :math:`\xi^e` maps
``v`` to a vector of subtype 0x42 or 0x44.
The function returns ``e`` if ``v`` is mapped to type 0x42
and ``0x100 + e`` if ``v`` is mapped to type 0x44. A negative
return value indicates that no such exponent ``e`` exists.
"""
assert v & 0x800 # Error if cocode part ov v is even
coc = (v ^ mat24.ploop_theta(v >> 12)) & 0xfff
# Obtain cocode as table of bit fields of 5 bits
tab = mat24.syndrome_table(coc & 0x7ff)
# Check if the syndrome bits are in 3 different MOG columns.
# We first XOR bit field i with bit field (i-1)(mod 3)
# and then zero the lowest two bits of each bit field.
tab ^= ((tab >> 5) & 0x3ff) ^ ((tab & 0x1f) << 10)
tab &= 0x739c
# Now all three bit fields are nonzero iff the syndrome bits
# are in three differnt columns. Next add 32 - 4 to each bit
# field in order to produce a carry if the field is nonzero.
tab += 0x739c
# Next we isolate the three carry bits
tab &= 0x8420
# Return -1 if all carry bits are set, i.e all syndrome bits
# are in different columns.
if (tab == 0x8420):
return -1
# Let scalar be the scalar product of the Golay part of v
# with the standard tetrad \omega
scalar = (v >> 22) & 1
# Exponent for element \xi of G_x0 is 2 - scalar
exp = 2 - scalar
if verbose:
w = gen_leech2_op_atom(v, 0x60000000 + exp)
print(
"Reducing c = %s, subtype %s, t=%s, e=%d, to v = %s, subtype %s" %
(hex(mat24.cocode_syndrome(coc, 0)), hex(gen_leech2_subtype(v)),
hex(tab), exp, hex(
mat24.gcode_to_vect(w >> 12)), hex(gen_leech2_subtype(w))))
# Return exponent for \xi in the lower 4 bits of the retrun value;
# Return 0 in bit 8 if all syndrome bits of v are in the same
# MOG column and 1 in bit 8 otherwise.
return ((tab != 0) << 8) + exp
def find_octad_permutation(v, result, verbose=0):
coc = (v ^ mat24.ploop_theta(v >> 12)) & 0xfff
w = mat24.gcode_weight(v >> 12)
vect = mat24.gcode_to_vect((v ^ ((w & 4) << 21)) >> 12)
src = mat24.vect_to_list(vect, 5)
syn = mat24.cocode_syndrome(coc, src[0]) & ~vect
if syn:
v5 = (1 << src[0]) | (1 << src[1]) | (1 << src[2])
v5 |= syn
special = mat24.syndrome(v5, 24)
src = src[:3]
src.append(mat24.lsbit24(special & vect))
src.append(mat24.lsbit24(vect & ~(special | v5)))
src.append(mat24.lsbit24(syn))
syn &= ~(1 << src[-1])
src.append(mat24.lsbit24(syn))
return apply_perm(v, src, OCTAD, len(src), result, verbose)
def syndrome_list(self, i=None):
"""Return syndrome of cocode element as list of bit positions.
:param i: ``None`` (default) or an integer ``0 <= i < 24``
used to select a syndrome.
:return: Syndrome of a Golay cocode element as a list
of at most four bit positions.
:raise:
* ValueError if argument ``i`` is ``None`` and the
syndrome is not unique.
The syndrome of the Golay cocode element is calculated
in the same way as in method **syndrome**. But here
the result is returned as an ordered list of bit positions
corresponding to the bits which are set in the syndrome.
"""
if i is None: i = 24
syn = mat24.cocode_syndrome(self.value, i)
return [i for i in range(24) if (syn >> i) & 1]
def leech2_start_type4(v):
"""Return subtype of a Leech lattice frame ``v`` used for reduction
The function returns the subtype of a vector ``v`` of type 4 in
the Leech lattice modulo 2. Parameter ``v2`` must be in Leech
lattice encoding. The function returns the subtype of ``v`` that
will be used for reduction in function ``gen_leech2_reduce_type4``.
In that function we take care of the special case that ``v + v0``
is of type 2 for a specific short vector ``v0``.
A simpler (but slower) implementation of thhis function is:
If ``v ^ v0`` is of type 2 the return the subtype of ``v ^ v0``.
Otherwise return the subtype of ``v``.
The function returns 0 if ``v`` is equal to ``Omega`` and
a negative value if ``v`` has not type 4.
This is a refernece implementation for function
``gen_leech2_start_type4()`` in file ``gen_leech.c``.
"""
if v & 0x7ff800 == 0:
# Then v or v + Omega is an even cocode element.
# Return 0 if v == Omega and -1 if v == 0.
if v & 0x7fffff == 0:
return 0 if v & 0x800000 else -1
# Let w be the cocode weight. Return -2 if w == 2.
if mat24.cocode_weight(v) != 4:
return -2
# Here v has type 4. Let v23 be the standard type-2 vector.
# Return 0x20 if v ^ v23 has type 2 and 0x40 otherwise.
return 0x20 if mat24.cocode_weight(v ^ 0x200) == 2 else 0x40
if mat24.scalar_prod(v >> 12, v):
# Then v has type 3 and we return -3
return -3
if v & 0x800:
# Then the cocode word 'coc' of v is odd.
coc = (v ^ mat24.ploop_theta(v >> 12)) & 0xfff
syn = mat24.cocode_syndrome(coc)
# If 'coc' has weight 1 then v is of type 2 and we return -2.
if (syn & (syn - 1)) == 0:
return -2
# Here v is of type 4.
# Return 0x21 if v ^ v23 is of type 2 and 0x43 otherwise.
if (syn & 0xc) == 0xc and (v & 0x200000) == 0:
return 0x21
return 0x43
# Let w be the weight of Golay code part divided by 4
w = mat24.gcode_weight(v >> 12)
if w == 3:
# Then the Golay code part of v is a docecad and we return 0x46.
return 0x46
# Here the Golay code part of v is a (possibly complemented) octad.
# Add Omega to v if Golay part is a complemented octad.
v ^= (w & 4) << 21
# Put w = 1 if that part is an octad and w = 0 otherwise.
w = (w >> 1) & 1
# Let 'octad' be the octad in the Golay code word in vector rep.
octad = mat24.gcode_to_vect(v >> 12)
coc = v ^ mat24.ploop_theta(v >> 12) # cocode element of v
# Return -2 if v is of type 2.
sub = suboctad_type(octad, w, coc)
if sub == 0:
return -2
# Return 0x22 if v ^ v23 is shsort
if suboctad_type(octad, w, coc ^ 0x200) == 0:
return 0x22
# Otherwise return the subtype of v
return 0x44 if sub & 2 else 0x42
def reduce_type4_std(v, verbose=0):
r"""Map type-4 vector in Leech lattice to standard vector
This is (almost) a python implementation of the C function
``gen_leech2_reduce_type4`` in file ``gen_leech.c``.
Let ``v \in \Lambda / 2 \Lambda`` of type 4 be given by
parameter ``v`` in Leech lattice encoding.
Let ``Omega`` be the type- vector in the Leech lattice
corresponding to the standard coordinate frame in the Leech
lattice.
Then the function constructs a ``g \in G_{x0}``
that maps ``v`` to ``Omega``.
The element ``g`` is returned as a word in the generators
of ``G_{x0}`` of length ``n \leq 6``. Each atom of the
word ``g`` is encoded as defined in the header
file ``mmgroup_generators.h``.
The function stores ``g`` as a word of generators in the
array ``pg_out`` and returns the length ``n`` of that
word. It returns a negative number in case of failure,
e.g. if ``v`` is not of type 4.
We remark that the C function ``gen_leech2_reduce_type4``
treats certain type-4 vectors ``v`` in a special way
as indicated in function ``reduce_type4``.
"""
if verbose:
print("Transforming type-4 vector %s to Omega" % hex(v & 0x1ffffff))
vtype = gen_leech2_subtype(v)
result = []
for _i in range(5):
coc = (v ^ mat24.ploop_theta(v >> 12)) & 0xfff
if verbose:
vt = gen_leech2_subtype(v)
coc_anchor = 0
if vt in [0x42, 0x44]:
w = mat24.gcode_weight(v >> 12)
vect = mat24.gcode_to_vect((v ^ ((w & 4) << 21)) >> 12)
coc_anchor = mat24.lsbit24(vect)
coc_syn = Cocode(coc).syndrome_list(coc_anchor)
gcode = mat24.gcode_to_vect(v >> 12)
print("Round %d, v = %s, subtype %s, gcode %s, cocode %s" %
(_i, hex(v & 0xffffff), hex(vt), hex(gcode), coc_syn))
assert vtype == gen_leech2_subtype(v)
if vtype == 0x48:
if verbose:
res = list(map(hex, result))
print("Transformation is\n", res)
return np.array(result, dtype=np.uint32)
elif vtype == 0x40:
if v & 0x7ffbff:
syn = mat24.cocode_syndrome(coc, 0)
src = mat24.vect_to_list(syn, 4)
v = apply_perm(v, src, LSTD, 4, result, verbose)
#print("after type 40", hex(v), Cocode(v).syndrome(0))
exp = 2 - ((v >> 23) & 1)
vtype = 0x48
elif vtype in [0x42, 0x44]:
exp = xi_reduce_octad(v)
if exp < 0:
v = find_octad_permutation(v, result, verbose)
exp = xi_reduce_octad(v)
assert exp >= 0
vtype = 0x40
elif vtype == 0x46:
exp = xi_reduce_dodecad(v, verbose)
if exp < 0:
vect = mat24.gcode_to_vect(v >> 12)
src = mat24.vect_to_list(vect, 4)
v = apply_perm(v, src, LSTD, len(src), result, verbose)
exp = xi_reduce_dodecad(v, verbose)
assert exp >= 0
vtype = 0x44
elif vtype == 0x43:
exp = xi_reduce_odd_type4(v, verbose)
if exp < 0:
vect = mat24.gcode_to_vect(v >> 12)
syn = mat24.cocode_syndrome(coc, 24)
src = mat24.vect_to_list(syn, 3)
#print("coc list", src)
v = apply_perm(v, src, LSTD[1:], len(src), result, verbose)
exp = xi_reduce_odd_type4(v, verbose)
assert exp > 0
vtype = 0x42 + ((exp & 0x100) >> 7)
exp &= 3
else:
raise ValueError("WTF")
if exp:
exp = 0xE0000003 - exp
v_old = v
v = gen_leech2_op_atom(v, exp)
assert v & 0xfe000000 == 0, (hex(v_old), hex(exp), hex(v))
result.append(exp)
raise ValueError("WTF1")
def reduce_type2_ortho(v, verbose=0):
r"""Map (orthgonal) short vector in Leech lattice to standard vector
This is a python implementation of the C function
``gen_leech2_reduce_type2_ortho`` in file ``gen_leech.c``.
Let ``v \in \Lambda / 2 \Lambda`` of type 2 be given by
parameter ``v`` in Leech lattice encoding.
In the real Leech lattice, (the origin of) the vector ``v`` must
be orthogonal to the standard short vector ``beta``. Here ``beta``
is the short vector in the Leech lattice propotional
to ``e_2 - e_3``, where ``e_i`` is the ``i``-th basis vector
of ``\{0,1\}^{24}``.
Let ``beta'`` be the short vector in the Leech lattice propotional
to ``e_2 + e_3``. Then the function constructs a ``g \in G_{x0}``
that maps ``v`` to ``beta'`` and fixes ``beta``.
The element ``g`` is returned as a word in the generators
of ``G_{x0}`` of length ``n \leq 6``. Each atom of the
word ``g`` is encoded as defined in the header
file ``mmgroup_generators.h``.
The function stores ``g`` as a word of generators in the
array ``pg_out`` and returns the length ``n`` of that
word. It returns a negative number in case of failure,
e.g. if ``v`` is not of type 2, or not orthogonal
to ``beta'`` in the real Leech lattice.
"""
vtype = gen_leech2_subtype(v)
if (vtype >> 4) != 2:
raise ValueError("Vector is not short")
if gen_leech2_type(v ^ 0x200) != 4:
raise ValueError("Vector not orthogonal to standard vector")
result = []
for _i in range(4):
if verbose:
coc = (v ^ mat24.ploop_theta(v >> 12)) & 0xfff
vt = gen_leech2_subtype(v)
coc_anchor = 0
if vt in [0x22]:
w = mat24.gcode_weight(v >> 12)
vect = mat24.gcode_to_vect((v ^ ((w & 4) << 21)) >> 12)
coc_anchor = mat24.lsbit24(vect)
coc_syn = Cocode(coc).syndrome_list(coc_anchor)
gcode = mat24.gcode_to_vect(v >> 12)
print("Round %d, v = %s, subtype %s, gcode %s, cocode %s" %
(_i, hex(v & 0xffffff), hex(vt), hex(gcode), coc_syn))
assert vtype == gen_leech2_subtype(v)
if vtype == 0x21:
exp = xi_reduce_odd_type2(v)
vtype = 0x22
elif vtype == 0x22:
exp = xi_reduce_octad(v)
if exp < 0:
w = mat24.gcode_weight(v >> 12)
vect = mat24.gcode_to_vect((v ^ ((w & 4) << 21)) >> 12)
if vect & 0x0c:
vect &= ~0x0c
src = mat24.vect_to_list(vect, 2) + [2, 3]
dest = [0, 1, 2, 3]
else:
src = [2, 3] + mat24.vect_to_list(vect, 3)
v5 = (1 << src[2]) | (1 << src[3]) | (1 << src[4])
v5 |= 0x0c
special = mat24.syndrome(v5, 24)
src.append(mat24.lsbit24(special & vect))
dest = [2, 3, 4, 5, 6, 7]
v = apply_perm(v, src, dest, len(src), result, verbose)
exp = xi_reduce_octad(v)
assert exp >= 0
vtype = 0x20
elif vtype == 0x20:
if ((v & 0xffffff) == 0x800200):
return np.array(result, dtype=np.uint32)
syn = (mat24.cocode_syndrome(v, 0)) & ~0xc
if syn and syn != 3:
src = mat24.vect_to_list(syn, 2) + [2, 3]
v = apply_perm(v, src, [0, 1, 2, 3], 4, result, verbose)
exp = 2 - ((v >> 23) & 1)
else:
raise ValueError("WTF")
if exp:
exp = 0xE0000003 - exp
v = gen_leech2_op_atom(v, exp)
result.append(exp)
raise ValueError("WTF1")
def reduce_type2(v, verbose=1):
r"""Map (orthgonal) short vector in Leech lattice to standard vector
This is a python implementation of the C function
``gen_leech2_reduce_type2`` in file ``gen_leech.c``.
Let ``v \in \Lambda / 2 \Lambda`` of type 2 be given by
parameter ``v`` in Leech lattice encoding.
Let ``beta`` be the short vector in the Leech lattice propotional
to ``e_2 - e_3``, where ``e_i`` is the ``i``-th basis vector
of ``\{0,1\}^{24}``.
Then the function constructs a ``g \in G_{x0}``
that maps ``v`` to ``beta``.
The element ``g`` is returned as a word in the generators
of ``G_{x0}`` of length ``n \leq 6``. Each atom of the
word ``g`` is encoded as defined in the header
file ``mmgroup_generators.h``.
The function stores ``g`` as a word of generators in the
array ``pg_out`` and returns the length ``n`` of that
word. It returns a negative number in case of failure,
e.g. if ``v`` is not of type 2.
"""
vtype = gen_leech2_subtype(v)
if (vtype >> 4) != 2:
raise ValueError("Vector is not short")
result = []
for _i in range(4):
if verbose:
coc = (v ^ mat24.ploop_theta(v >> 12)) & 0xfff
vt = gen_leech2_subtype(v)
coc_anchor = 0
if vt in [0x22]:
w = mat24.gcode_weight(v >> 12)
vect = mat24.gcode_to_vect((v ^ ((w & 4) << 21)) >> 12)
coc_anchor = mat24.lsbit24(vect)
coc_syn = Cocode(coc).syndrome_list(coc_anchor)
gcode = mat24.gcode_to_vect(v >> 12)
print("Round %d, v = %s, subtype %s, gcode %s, cocode %s" %
(_i, hex(v & 0xffffff), hex(vt), hex(gcode), coc_syn))
assert vtype == gen_leech2_subtype(v)
if vtype == 0x21:
exp = xi_reduce_odd_type2(v)
vtype = 0x22
elif vtype == 0x22:
exp = xi_reduce_octad(v)
if exp < 0:
w = mat24.gcode_weight(v >> 12)
vect = mat24.gcode_to_vect((v ^ ((w & 4) << 21)) >> 12)
src = mat24.vect_to_list(vect, 4)
dest = [0, 1, 2, 3]
v = apply_perm(v, src, dest, 4, result, verbose)
exp = xi_reduce_octad(v)
assert exp >= 0
vtype = 0x20
elif vtype == 0x20:
exp = 0
# map v to stadard cocode word [2,3]
if v & 0x7fffff != 0x200:
syn = (mat24.cocode_syndrome(v, 0))
src = mat24.vect_to_list(syn, 2)
v = apply_perm(v, src, [2, 3], 2, result, verbose)
# correct v2 if v2 is the cocode word [2,3] + Omega
if v & 0x800000:
atom = 0xC0000200
# operation y_d such that d has odd scalar
# product with cocode word [2,3]
v = gen_leech2_op_atom(v, atom)
result.append(atom)
assert v & 0xffffff == 0x200
return np.array(result, dtype=np.uint32)
else:
raise ValueError("WTF")
if exp:
exp = 0xE0000003 - exp
v = gen_leech2_op_atom(v, exp)
result.append(exp)
raise ValueError("WTF1")
