Recursive Structure of Hulls of PRM Codes

TL;DR

This paper derives a recursive formula for the hull dimensions of projective Reed-Muller codes using combinatorial numbers, applicable within a specific parameter range, and extends results via duality.

Contribution

It introduces a new recursive structure for hulls of PRM codes and provides explicit formulas for their dimensions in certain ranges.

Findings

Derived a recursive formula for hull dimensions of PRM codes.

Provided explicit formulas for hull dimensions within the open lower-half range.

Extended results to the upper range using duality.

Abstract

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}.