A Matching Vector ($\mathbf{MV}$) family modulo a positive integer $m\ge 2$ is a
pair of ordered lists $U=(u_1,\ldots,u_K)$ and $V=(v_1,\ldots,v_K)$ with
$u_i,v_j\in \Z_m^n$ such that $\langle u_i,v_i\rangle=0 \pmod m$ for every
$i\in[K]$, while $\langle u_i,v_j\rangle\ne 0 \pmod m$ for every $i\ne j$. It
is called $r$-restricted if the set of all inner products
$\langle u_i,v_j\rangle$ has size at most $r$, including the diagonal value
$0$. Restricted matching vector families are central to the matching-vector
construction of constant-query subexponential locally decodable codes: an
$r$-restricted family of size $K$ in $\Z_m^n$ gives an $r$-query matching-vector
code with message length $K$ and codeword length $N=m^n$.

The best previous bound tailored to the $3$-restricted case was
$MV(m,n,3)\le \exp\bigl(O_m(n/\log n)\bigr)$ for fixed $m$, due to Bhowmick, Dvir and Lovett~\cite{bdl13}, who showed this bound under the polynomial Freiman Ruzsa conjecture, which was proved by Gowers, Green, Manners, and Tao~\cite{PFR24}. In this
work we prove the subexponential upper bound
\[
MV(m,n,3) \le \exp\bigl(O_m(\sqrt{n\log n})\bigr)
\]
for every fixed modulus $m$. Consequently, every $3$-query matching-vector
code over a fixed modulus has codeword length
\[
N \ge \exp\left(\Omega_m\left(\frac{(\log K)^2}{\log\log K}\right)\right).
\]
Our proof answers an explicit technical question left by the work of Aggarwal, Dutta, Li, Obremski, and Saraogi~\cite{ADLOS}: the
entropy-growth method for sums of matching vectors can be pushed far beyond the
initial constant-sum regime. We show that, for $3$-restricted families, entropy
continues to grow for sums of length $L$ as large as
$\Theta_m(\sqrt{n/\log n})$. The new ingredient is an exact-count rank
bootstrap: a long-sum collision changes the exact number of one residue class
seen from a suitable pivot, and exact-weight polynomials convert a disjoint
collision family into a low-rank identity matrix.