def gen_xi_short_to_leech(x1):
"""Convert short vector to Leech lattice encoding.
Both, Leech lattice and short vector encoding of a short vector
in Q_x are decribed in the header of this module. The function
returns the Leech lattice encoding of element x1 given in short
vector encoding.
The function returns 0 for an illegal input x1.
"""
box, sign, code = x1 >> 16, (x1 >> 15) & 1, x1 & 0x7fff
octad = 0xffff
if box == 1:
if code < 1536: # this is 2 * 24 * 32
gcode = code >= 768
if gcode:
code -= 768
gcode <<= 11
i, j = code >> 5, code & 31
cocode = Mat24.vect_to_cocode((1 << i) ^ (1 << j))
if cocode == 0 or cocode & 0x800:
return 0
elif code < 2496: # this is 2 * 24 * 32 + 15 * 64
octad = code - 1536
else:
return 0
elif box == 2:
if code >= 23040: # this is 360 * 64
return 0
octad = code + 960 # this is 15 * 64
elif box == 3:
if code >= 24576: # this is 384 * 64
return 0
octad = code + 24000 # this is (15 + 360) * 64
elif box < 6:
code += (box - 4) << 15
cocode = Mat24.vect_to_cocode(1 << (x1 & 31))
if cocode == 0:
return 0
gcode = (code >> 5) & 0x7ff
w = Mat24.gcode_weight(gcode) ^ Mat24.scalar_prod(gcode, cocode)
gcode ^= (w & 1) << 11
else:
return 0
if octad < 48756: # this is 759 * 64
cc = octad & 0x3f
w = Mat24.bw24(cc)
cc = (cc << 1) ^ (w & 1)
w += w & 1
gcode = Mat24.octad_to_gcode(octad >> 6)
gcodev = Mat24.gcode_to_vect(gcode)
cocode = Mat24.vect_to_cocode(Mat24.spread_b24(cc, gcodev))
gcode ^= ((w >> 1) & 1) << 11
cocode ^= Mat24.ploop_theta(gcode)
return (sign << 24) | (gcode << 12) | cocode
def w2_gamma(v):
"""Implement function gamma() in [Seysen20], section 3.3.
Given a Golay code vector v in 'gcode' representation, see
module mat24.py, we return a pair w2, c. Here c is equal to
the element gamma(v) of the Golay cocode, with gamma() as in
[Seysen20], section 3.3. ,c is returned in 'cocode' representation,
see module mat24.py. We also return the bit w2 = w2(c), with w2()
as defined in [Seysen20], section 3.3.
"""
x = Mat24.gcode_to_vect(v)
x1 = sum( (x >> i) & 0x111111 for i in range(1,4) )
x1 = (x1 >> 1) & 0x111111
w2 = bw24(x1)
w2 = ((w2 * (w2 - 1)) >> 1) & 1
c = Mat24.vect_to_cocode(x1)
return w2, c
def y_table(g, cocodes, sign, code_omega, cocode_x, verbose = 0):
"""Return table for function ``gen_leech2_reduce_n``
This table is used to map an entry :math:`x_d x_\delta`,
where :math:`d` is a (signed) octad or dodecad, and
where :math:`\delta` is even, to a standard form. The
table is to be used for a fixed Golay code element
:math:`d` up to sign and up to a factor :math:`\Omega`.
Such a mapping is done via conjugation with an element
of shape :math:`y_e x_\epsilon`, with math:`e` a Goly
code word math:`\epsilon` an even cocode word..
Here :math:`d` is given by parameter ``g``. Thee we choose
a Golay code word ``e`` with :math:`g \cap e = \delta`.
Using the returned table, ``e`` may be calculated as
:math:`e = \sum_{i=0}^10 \delta_i \cdot t_i` ,
where :math:`\delta_i` is the ``i``-th bit of
:math:`\delta`, and :math:`t_i` is the entry ``table[i+1]``
of the table.
``table[0]`` is equal to :math:`\theta(d)`. ``table[12]``
is a sign bit ``s``. If the bit 23 of :math:`v \cdot y_e`
differs from ``s`` then we have to replace :math:`e` by
:math:`e + e'`, where :math:`e'` is given by ``table[13]``.
This operation adds :math:`x_\Omega` to :math:`x_d` in
the cases relevant for function ``gen_leech2_reduce_n``.
If :math:`v \cdot y_{e + e'}` has a sign bit set then
we have to multiply that vector with :math:`x_\epsilon`,
where the cocode word :math:`epsilon` is given by
``table[13]``. Entries ``table[11,12,13]`` are copied
from parameters ``sign, code_omega, cocode_x``,
respectively.
Parameter ``cocode_x`` is an list of cocode words
that will be meapped to zero by applying the table.
On input, all Golay code and cocode word must be given
as list of integers representing the bit positions set.
In the returned table, all Golay code or cocode words
are given as bit vector in **Golay code** or **cocode**
representation, respectively.
"""
gv = Mat24.vect_to_gcode(list_to_vect(g))
b = []
for i in range(11):
b.append((Mat24.ploop_cap(gv, 1 << i) << 12) + (1 << i))
for c in cocodes:
b.append(Mat24.vect_to_cocode(list_to_vect(c)) << 12)
if verbose:
print("Function y_table, g =", hex(list_to_vect(g)))
print("cocodes =", cocodes)
print("cocode_x =", cocode_x)
print("b =", [hex(x) for x in b])
b, columns = pivot_binary_high(b)
if verbose:
print("reduced b =", [hex(x) for x in b])
print("columns =", columns)
table = [Mat24.ploop_theta(gv)]
table1 = []
for i in range(10, -1, -1):
if columns[0] == i + 12:
table1.append(b[0] & 0xfff)
b, columns = b[1:], columns[1:]
else:
table1.append(0)
table1.reverse()
table += table1
table.append(sign)
table.append(Mat24.vect_to_gcode(list_to_vect(code_omega)))
table.append(Mat24.vect_to_cocode(list_to_vect(cocode_x)))
if verbose:
print("table =", [hex(x) for x in table])
return table
def vector(gcode, cocode):
gc = Mat24.vect_to_gcode(sum(1 << x for x in gcode))
coc = Mat24.vect_to_cocode(sum(1 << x for x in cocode))
return (gc << 12) ^ coc ^ Mat24.ploop_theta(gc)