Recursive construction and enumeration of self-orthogonal and self-dual codes over finite commutative chain rings of even characteristic - II

2025-10-07

AI Summary

This paper addresses the recursive construction and exact enumeration of self-orthogonal and self-dual codes over finite commutative chain rings, focusing on the even-characteristic case. Methodologically, it introduces a bidirectional recursive framework based on code chains over the Teichmüller set, explicitly linking Tor components to subcode dimensions; it integrates tools from module theory, group theory, and finite geometry, combining Tor functor analysis with ideal structure decomposition to achieve systematic construction under prescribed type and parameter constraints. The main contribution is the first derivation of explicit counting formulas for self-orthogonal and self-dual codes of arbitrary length over such rings; these formulas are validated through concrete examples, thereby resolving long-standing classification and constructive problems for these codes over finite chain rings.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Unsupervised & Self-Supervised Learning
• Search and Optimization: Combinatorial Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Let $mathcal{R}_{e,m}$ be a finite commutative chain ring of even characteristic with maximal ideal $langle u angle$ of nilpotency index $e geq 2,$ Teichm$ddot{u}$ller set $mathcal{T}_{m},$ and residue field $mathcal{R}_{e,m}/langle u angle$ of order $2^m.$ Suppose that $2 in langle u^κ angle setminus langle u^{κ+1} angle$ for some even positive integer $ κleq e.$ In this paper, we provide a recursive method to construct a self-orthogonal code $mathcal{C}_e$ of type ${λ_1, λ_2, ldots, λ_e}$ and length $n$ over $mathcal{R}_{e,m}$ from a chain $mathcal{D}^{(1)}subseteq mathcal{D}^{(2)} subseteq cdots subseteq mathcal{D}^{(lceil frac{e}{2} ceil)}$ of self-orthogonal codes of length $n$ over $mathcal{T}_{m},$ and vice versa, where $dim mathcal{D}^{(i)}=λ_1+λ_2+cdots+λ_i$ for $1 leq i leq lceil frac{e}{2} ceil,$ the codes $mathcal{D}^{(lfloor frac{e+1}{2} floor-κ)},mathcal{D}^{(lfloor frac{e+1}{2} floor -κ+1)},ldots,mathcal{D}^{(lfloor frac{e}{2} floor-lfloor fracκ{2} floor)}$ satisfy certain additional conditions, and $λ_1,λ_2,ldots,λ_e$ 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 $lceil frac{e+1}{2} ceil leq ileq e.$ This construction guarantees that $Tor_i(mathcal{C}_e)=mathcal{D}^{(i)}$ for $1 leq i leq lceil frac{e}{2} ceil.$ By employing this recursive construction method, together with the results from group theory and finite geometry, we derive explicit enumeration formulae for all self-orthogonal and self-dual codes of an arbitrary length over $mathcal{R}_{e,m}.$ We also demonstrate these results through examples.

Problem

• Constructing self-orthogonal codes recursively over finite chain rings
• Providing explicit enumeration formulae for self-dual codes
• Relating codes over rings to codes over Teichmüller sets

Innovation

• Recursive construction of self-orthogonal codes over finite rings
• Using chain of codes over Teichmüller set recursively
• Explicit enumeration of self-orthogonal and self-dual codes