Recursive construction and enumeration of self-orthogonal and self-dual codes over finite commutative chain rings of even characteristic
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866911376505569280 |
|---|---|
| author | Yadav, Monika Sharma, Anuradha |
| author_facet | Yadav, Monika Sharma, Anuradha |
| contents | Let $\mathscr{R}_{e,m}$ denote a finite commutative chain ring of even characteristic with maximal ideal $\langle u \rangle$ of nilpotency index $e \geq 3,$ Teichm$\ddot{u}$ller set $\mathcal{T}_{m},$ and residue field $\mathscr{R}_{e,m}/\langle u \rangle$ of order $2^m.$ Suppose that $2 \in \langle u^κ\rangle \setminus \langle u^{κ+1}\rangle$ for some odd integer $κ$ with $3 \leq κ\leq e.$ In this paper, we first develop a recursive method to construct a self-orthogonal code $\mathscr{D}_e$ of type $\{λ_1, λ_2, \ldots, λ_e\}$ and length $n$ over $\mathscr{R}_{e,m}$ from a chain $\mathcal{C}^{(1)}\subseteq \mathcal{C}^{(2)} \subseteq \cdots \subseteq \mathcal{C}^{(\lceil \frac{e}{2} \rceil)} $ of self-orthogonal codes of length $n$ over $\mathcal{T}_{m},$ and vice versa, subject to certain conditions, where $λ_1,λ_2,\ldots,λ_e$ are non-negative integers satisfying $2λ_1+2λ_2+\cdots+2λ_{e-i+1}+λ_{e-i+2}+λ_{e-i+3}+\cdots+λ_i \leq n$ for $\lceil \frac{e+1}{2} \rceil \leq i\leq e,$ and
$\lfloor \cdot \rfloor$ and $\lceil \cdot \rceil$ denote the floor and ceiling functions, respectively. This construction ensures that $Tor_i(\mathscr{D}_e)=\mathcal{C}^{(i)}$ for $1 \leq i \leq \lceil \frac{e}{2} \rceil.$
With the help of this recursive construction method and by applying results from group theory and finite geometry, we obtain explicit enumeration formulae for all self-orthogonal and self-dual codes of an arbitrary length over $\mathscr{R}_{e,m}.$ We also illustrate these results with some examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_06082 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Recursive construction and enumeration of self-orthogonal and self-dual codes over finite commutative chain rings of even characteristic Yadav, Monika Sharma, Anuradha Information Theory 15A63, 94B99, 94B15 Let $\mathscr{R}_{e,m}$ denote a finite commutative chain ring of even characteristic with maximal ideal $\langle u \rangle$ of nilpotency index $e \geq 3,$ Teichm$\ddot{u}$ller set $\mathcal{T}_{m},$ and residue field $\mathscr{R}_{e,m}/\langle u \rangle$ of order $2^m.$ Suppose that $2 \in \langle u^κ\rangle \setminus \langle u^{κ+1}\rangle$ for some odd integer $κ$ with $3 \leq κ\leq e.$ In this paper, we first develop a recursive method to construct a self-orthogonal code $\mathscr{D}_e$ of type $\{λ_1, λ_2, \ldots, λ_e\}$ and length $n$ over $\mathscr{R}_{e,m}$ from a chain $\mathcal{C}^{(1)}\subseteq \mathcal{C}^{(2)} \subseteq \cdots \subseteq \mathcal{C}^{(\lceil \frac{e}{2} \rceil)} $ of self-orthogonal codes of length $n$ over $\mathcal{T}_{m},$ and vice versa, subject to certain conditions, where $λ_1,λ_2,\ldots,λ_e$ are non-negative integers satisfying $2λ_1+2λ_2+\cdots+2λ_{e-i+1}+λ_{e-i+2}+λ_{e-i+3}+\cdots+λ_i \leq n$ for $\lceil \frac{e+1}{2} \rceil \leq i\leq e,$ and $\lfloor \cdot \rfloor$ and $\lceil \cdot \rceil$ denote the floor and ceiling functions, respectively. This construction ensures that $Tor_i(\mathscr{D}_e)=\mathcal{C}^{(i)}$ for $1 \leq i \leq \lceil \frac{e}{2} \rceil.$ With the help of this recursive construction method and by applying results from group theory and finite geometry, we obtain explicit enumeration formulae for all self-orthogonal and self-dual codes of an arbitrary length over $\mathscr{R}_{e,m}.$ We also illustrate these results with some examples. |
| title | Recursive construction and enumeration of self-orthogonal and self-dual codes over finite commutative chain rings of even characteristic |
| topic | Information Theory 15A63, 94B99, 94B15 |
| url | https://arxiv.org/abs/2510.06082 |