A family of m-ovoids of parabolic quadrics

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Problem: Reconstructed conjecture: for every prime power $q$, every $m$-ovoid $\mathcal M$ of the parabolic quadric $Q(4,q)\subseteq \mathrm{PG}(4,q)$, and every elliptic hyperplane section $E\cong Q^-(3,q)$, one has

Here an $m$-ovoid means a point set meeting every generator line of $Q(4,q)$ in exactly $m$ points. This matches the paper’s Remark 4.1 wording.

Result: The conjecture is false. A counterexample occurs for $q=4$, $m=2$.

Let $\mathbb F_4=\mathbb F_2(\omega)$, $\omega^2+\omega+1=0$, and

Let

For $P\in Q\setminus E$, scale its representative uniquely so that $\ell(P)=1$, say $P=(a,b,c,d,e)$, and define

Let $\operatorname{Tr}(z)=z+z^2$, and set

The following exhaustive certificate over $\mathbb F_4$ verifies all claims.

from itertools import product, combinations
MOD=0b111; QN=4; w=2; w2=3

def mul(a,b):
    r=0; aa=a; bb=b
    while bb:
        if bb&1: r ^= aa
        bb >>= 1; aa <<= 1
        if aa & QN: aa ^= MOD
    return r & 3
def sq(a): return mul(a,a)
def inv(a): return 1 if a==1 else (3 if a==2 else 2)
def tr(a): return a ^ sq(a)
def smul(c,v): return tuple(mul(c,x) for x in v)
def vadd(u,v): return tuple(x^y for x,y in zip(u,v))
def norm(v):
    for x in v:
        if x: return smul(inv(x),v)
    raise ValueError

def qform(v): return mul(v[0],v[1]) ^ mul(v[2],v[3]) ^ sq(v[4])
def polar(u,v): return mul(u[0],v[1]) ^ mul(u[1],v[0]) ^ mul(u[2],v[3]) ^ mul(u[3],v[2])
def ell(v): return v[2] ^ v[3] ^ mul(w,v[4])
def Phi(v):
    a,b,c,d=v[:4]
    return mul(w2,sq(a)) ^ mul(w,mul(a,b)) ^ mul(a,c) ^ mul(w2,sq(b)) ^ \
           mul(w2,mul(b,c)) ^ mul(w2,mul(b,d)) ^ sq(c) ^ mul(c,d)

def inM(P):
    L=ell(P)
    if L==0: return False
    V=smul(inv(L),P)
    return tr(Phi(V))==1

pts=[]
for v in product(range(4), repeat=5):
    if v!=(0,0,0,0,0) and qform(v)==0:
        P=norm(v)
        if P not in pts: pts.append(P)

lines=set()
for P,R in combinations(pts,2):
    if polar(P,R)==0:
        L=frozenset(norm(vadd(smul(a,P),smul(b,R)))
                    for a in range(4) for b in range(4) if (a,b)!=(0,0))
        lines.add(L)

E={P for P in pts if ell(P)==0}
M={P for P in pts if inM(P)}

assert len(pts)==85
assert len(lines)==85
assert len(E)==17
assert all(not L <= E for L in lines)   # E is elliptic
assert len(M)==34
assert all(len(L & M)==2 for L in lines)
assert len(E & M)==0

Thus $E\cong Q^-(3,4)$, $\mathcal M$ is a $2$-ovoid of $Q(4,4)$, but

So the conjectured congruence modulo $q$ fails.

Citation: No published counterexample is used here. Context: T. Feng, K. Momihara, Q. Xiang, “A family of $m$-ovoids of parabolic quadrics,” J. Combin. Theory Ser. A 140 (2016), Remark 4.1.