Recursive Structure of Hulls of PRM Codes
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908989763092480 |
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| author | Song, Yufeng Yue, Qin |
| author_facet | Song, Yufeng Yue, Qin |
| contents | 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}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21808 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Recursive Structure of Hulls of PRM Codes Song, Yufeng Yue, Qin Information Theory 11T71, 94B05 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}. |
| title | Recursive Structure of Hulls of PRM Codes |
| topic | Information Theory 11T71, 94B05 |
| url | https://arxiv.org/abs/2604.21808 |