Recursive Structure of Hulls of PRM Codes

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: $$\dim \Hull\bigl(\PRM(q,m,v)\bigr) = \dim \PRM(q,m,v) - \sum_{i=0}^{\ell}A_{2i+\epsilon}\bigl(v-(\ell-i)(q-1)\bigr),$$ where $$\ell=\Bigl\lfloor\frac r2\Bigr\rfloor,\qquad \epsilon= \begin{cases} 0, & r\ \text{is even}, 1, & r\ \text{is odd}. \end{cases}$$ 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 S\o rensen's duality theorem \cite{Sorensen}.