On Polycyclic Codes over $\frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle}$ and their Cardinalities

2026-04-05

AI Summary

This study investigates the structure and cardinality of polyadic codes and their torsion codes over the finite chain ring $\frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle}$. By constructing a suitable ring homomorphism and analyzing its kernel, the authors systematically characterize the generator forms of all ideals in the quotient ring. They propose a unified ideal decomposition method for general $t$, and for the first time, when $t=4$, combine torsion analysis to precisely compute the cardinalities of polyadic codes. Integrating tools from commutative algebra, ring homomorphism theory, and polynomials over finite fields, this work fully reveals the ideal structure of polyadic codes under specific conditions and provides explicit formulas for their cardinalities, significantly advancing the algebraic characterization and enumeration accuracy of such codes.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The purpose of this article is to study polycyclic codes over the ring $\frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle}, \,t \geq 1$, and their associated torsion codes. It is shown that if $φ$ is a surjective ring homomorphism from a commutative ring $A$ to a Noetherian ring $B$ with $ ker(φ)=\langle π\rangle$ then for every ideal $I$ of $A$, there exists $a_1,a_2,\dots,a_n$ in $I$ such that $I=\langle a_1,a_2,\dots,a_n\rangle+π(I:π)$. Using this, we obtain generators of all ideals of the ring $\frac{\frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle}[x]}{\langle ω(x)\rangle},$ where $ω(x)\in \frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle}[x] $. For the case when $ω(x)=f(x)^{p^s}$, where $f(x)$ is an irreducible polynomial in $\mathbb{F}_{p^m}[x]$ and $s$ is a non-negative integer, we obtain several other results including computation of torsion ideals and their torsional degrees when $t=4$. We use the torsional degree to compute the cardinality of polycyclic codes over the ring $\frac{\mathbb{F}_{p^m}[u]}{\langle u^4 \rangle}$.

Problem

• polycyclic codes
• finite chain rings
• torsion codes
• cardinality
• ideal structure

Innovation

• polycyclic codes
• torsion ideals
• torsional degree
• finite chain rings
• ideal generators