A Study Of Skew-Polycyclic Codes Over A Non-Chain Ring

Jul 16, 2026

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A study of skew-polycyclic codes built from skew polynomial rings over a finite non-chain ring Ru2,v2,pm constructed from the finite field Fpm. For a central polynomial f(x) and an automorphism Θ, the work describes skew polycyclic codes of length lj tied to f(x)^j, characterizes free codes, and computes their ranks. With centrality conditions, it decomposes quotient rings for xnps−λ (gcd(n,p)=1, Θ(λ)=λ) to analyze skew constacyclic codes via left ideals, then illustrates optimality with examples and extends prior constacyclic results.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages18

Reading Time45 min

Abstract

Introduction

arXiv :2607 .08304v 1 [ cs .IT] 9 Jul 2026 A STUDY OF SKEW-POLYCYCLIC CODES OVER A NON-CHAIN RING SEEMA ANTIL, SEEMA CHAHAL, MANJU KHAN, AND SUGANDHA MAHESHWARY Abstract. For a prime p and a positive integer m, let Fpm be the finite field of cardinality pm , and let Ru2 ,v2 ,pm = Fpm + uFpm + vFpm + uvFpm , u2 = v 2 = 0, uv = vu, be a finite non-chain ring. In this paper, we study skew polycyclic codes of length lj associated with f (x)j , where f (x) is a central polynomial of degree l in Ru2 ,v2 ,pm [x;Θ], where Θ being an automorphism of Ru2 ,v2 ,pm . We describe these codes, characterize free skew polycyclic codes, and determine their ranks. Under suitable centrality assumptions, we decompose the quotient ring associated with xnps − λ, where gcd(n,p) = 1 and Θ(λ) = λ . This reduces the study of skew (λ,Θ)-constacyclic codes of length nps to the study of left ideals of Ru2~~ ~~,⟨vf2(,pxm)~~j~~⟩[x;Θ], where f (x) is a central irreducible divisor of degree l of xnps − λ, for an invertible element λ ∈ Ru2 ,v2 ,pm and j ∈ N. We then apply these results to skew (λ,Θ)-constacyclic codes of length ps for different classes of units λ . Several examples are presented to illustrate the theory and to obtain optimal codes. Finally, when Θ is the identity automorphism, we study constacyclic codes of length nps over Ru2 ,v2 ,pm , according as xn − α0 is irreducible or reducible over Fpm . These results extend the work of [CCD+ 18] and [ZTG18] on constacyclic codes of length nps over Fpm +uFpm to the finite non-chain ring Ru2 ,v2 ,pm . 1. Introduction The class of constacyclic codes has attracted considerable attention due to its rich algebraic structure and numerous applications in coding theory. In particular, the classification of such codes plays a crucial role in understanding their algebraic structure. In the last two decades, considerable attention has been devoted to studying constacyclic codes over various finite rings. These codes can be utilized in cryptography, data transmission, data compression, and storage systems, where they play an important role in the detection and correction of errors in various communication channels. Although extensive research has been carried out on constacyclic codes over finite rings, a complete classification is generally difficult and is known only for certain lengths over specific finite fields and finite chain rings. In this context, Zhao et al. [ZTG18] and Cao et al.[CCD+18] determined the constacyclic codes of length nps over Fpm + uFpm . Also A noncommutative generalization of cyclic and constacyclic codes is obtained by using skew polynomial rings. Boucher et al. [BGU07] introduced skew cyclic codes over finite fields by considering the skew polynomial ring Fpm [x;θ], where θ is an automorphism of Fpm . Later, skew constacyclic and skew polycyclic codes were studied over several classes of finite rings; see [JLU12, HS23 , RPM26 , CAMK26 , TS26 , BMMOa26] . Since skew polynomial rings are usually non-commutative, their factorization theory is richer than that of ordinary polynomial rings. This provides a useful framework for obtaining new families of codes. Beyond finite chain rings, researchers have also studied codes over finite non-chain rings. Yildiz and Karadeniz [YK11] studied cyclic codes over the non-chain ring F2 +uF2 +vF2 +uvF2 , where u2 = v 2 = 0, uv = vu. Dougherty et al. , [DKY12] studied cyclic codes over the family of rings Rk = F2 [u1 ,..., uk]/⟨u2i, uiuj − ujui | 1 ≤ i, j ≤ k⟩, which includes non-chain rings for 2010 Mathematics Subject Classification. 16S36, 94B05, 94B15, 94B60 . Key words and phrases. finite non-chain ring, skew polynomials, skew constacyclic codes, repeated root codes. The first-named author gratefully acknowledges the financial support provided by the Ministry of Education, Government of India. The first author also acknowledges the partial support from the FIST program of the Department of Science and Technology, Government of Ind

FAQs

What ring and algebraic setting does the paper use for skew-polycyclic codes?

It uses the finite non-chain ring Ru2,v2,pm built from Fpm with u^2=v^2=0 and uv=vu, together with a skew polynomial ring Ru2,v2,pm[x;Θ] where Θ is an automorphism satisfying xa=Θ(a)x.

How are the skew polycyclic codes in this work defined?

For j∈N and a central polynomial f(x) in Ru2,v2,pm[x;Θ], the skew fj,Θ-polycyclic codes correspond to left ideals of a quotient ring associated with a polynomial module construction driven by f(x)^j.

How does the paper reduce the study of skew constacyclic codes?

Under suitable centrality assumptions with gcd(n,p)=1 and Θ(λ)=λ, it decomposes the quotient ring for xnps−λ so that skew (λ,Θ)-constacyclic codes of length nps reduce to studying left ideals tied to skew (fj,Θ)-polycyclic codes of length jl.

Jul 16, 2026

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