For a nonnegative integer $r$ and a positive integer $v$ satisfying \[ \frac{r(q-1)}{2}<v<\frac{(r+1)(q-1)}{2}, \] we define the combinatorial numbers \[ A_r(v)= \begin{cases} \displaystyle \sum_{t=r(q-1)-v}^{v}\ \sum_{j=0}^{r}(-1)^j\binom{r}{j}\binom{t-jq+r-1}{r-1}, & r>0,\\[1.2ex] 1, & r=0. \end{cases} \] For the projective Reed-Muller code $\PRM(q,m,v)$, we determine its hull dimension: \[ \dim \Hull\bigl(\PRM(q,m,v)\bigr) = \dim \PRM(q,m,v) - \sum_{i=0}^{\ell}A_{2i+ε}\bigl(v-(\ell-i)(q-1)\bigr), \] where \[ \ell=\Bigl\lfloor\frac r2\Bigr\rfloor,\qquad ε= \begin{cases} 0, & r\ \text{is even}, 1, & r\ \text{is odd}. \end{cases} \] This formula applies in the open lower-half range $ 0<v<\frac{m\Qm}{2}, $ equivalently for $v\in I_r$ with $m\ge r+1$; the range $ \frac{m\Qm}{2}<v<m\Qm $ is then obtained by Sørensen's duality theorem \cite{Sorensen}.

翻译：对于满足 \[ \frac{r(q-1)}{2}<v<\frac{(r+1)(q-1)}{2} \] 的非负整数 $r$ 和正整数 $v$，我们定义组合数 \[ A_r(v)= \begin{cases} \displaystyle \sum_{t=r(q-1)-v}^{v}\ \sum_{j=0}^{r}(-1)^j\binom{r}{j}\binom{t-jq+r-1}{r-1}, & r>0,\\[1.2ex] 1, & r=0. \end{cases} \] 对于射影Reed-Muller码 $\PRM(q,m,v)$，我们确定其壳维度： \[ \dim \Hull\bigl(\PRM(q,m,v)\bigr) = \dim \PRM(q,m,v) - \sum_{i=0}^{\ell}A_{2i+ε}\bigl(v-(\ell-i)(q-1)\bigr), \] 其中 \[ \ell=\Bigl\lfloor\frac r2\Bigr\rfloor,\qquad ε= \begin{cases} 0, & r\ \text{为偶数}, 1, & r\ \text{为奇数}. \end{cases} \] 此公式适用于开下半区间 $ 0<v<\frac{m\Qm}{2} $，即当 $m\ge r+1$ 时 $v\in I_r$；区间 $ \frac{m\Qm}{2}<v<m\Qm $ 则由Sørensen对偶定理 \cite{Sorensen} 得到。