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

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 of even characteristic. Methodologically, leveraging the Teichmüller structure of such rings, we introduce a bidirectional recursive framework based on code-chain lifting—establishing, for the first time in even characteristic, a bijective correspondence between self-orthogonal codes over the ring and their underlying Teichmüller code chains, while uncovering intrinsic links between Tor components and chain structures. Combining ring-theoretic analysis, Tor functor techniques, orbit counting under group actions, and finite geometric methods, we derive explicit closed-form enumeration formulas for self-orthogonal and self-dual codes of arbitrary length. The theoretical results are validated through concrete examples, ensuring both mathematical rigor and computational feasibility. This work provides a systematic, unified paradigm for code construction over finite chain rings.

Technology Category

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

Application Category

• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Let $mathscr{R}_{e,m}$ denote a finite commutative chain ring of even characteristic with maximal ideal $langle u angle$ of nilpotency index $e geq 3,$ Teichm$ddot{u}$ller set $mathcal{T}_{m},$ and residue field $mathscr{R}_{e,m}/langle u angle$ of order $2^m.$ Suppose that $2 in langle u^κ angle setminus langle u^{κ+1} angle$ for some odd integer $κ$ with $3 leq κleq e.$ 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} ceil)} $ 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 $lceil frac{e+1}{2} ceil leq ileq e,$ and $lfloor cdot floor$ and $lceil cdot ceil$ denote the floor and ceiling functions, respectively. This construction ensures that $Tor_i(mathscr{D}_e)=mathcal{C}^{(i)}$ for $1 leq i leq lceil frac{e}{2} ceil.$ 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.

Problem

• Construct self-orthogonal codes over finite commutative chain rings recursively
• Develop enumeration formulae for self-orthogonal and self-dual codes
• Establish relationships between codes over rings and Teichmüller sets

Innovation

• Recursive construction of self-orthogonal codes over finite rings
• Using chain of codes over Teichmüller set with conditions
• Explicit enumeration formulae for self-dual codes via group theory