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International Conference on Advanced Communication Technologies and Networking (CommNet)

Quantum and LCD Codes from the Consatacyclic Codes derived from the Ring R

Mohammed Sabiri, Bassou Aouijil

Published Dec 3, 2025• 0 citations
Publisher

Abstract

Let${\mathcal{R}} = {{\mathbb{F}}_q}[u,v]/,$ where ${{\mathbb{F}}_q}$ is a finite of cardinality q and u2= u,v3= v,uv = vu. In this work, we investigate quantum and LCD codes over ${{\mathbb{F}}_q}$ by using constacyclic codes over the ring ${\mathcal{R}}$, for the first codes, dual or not dual containing codes were used, and for the second codes, constacylic LCD codes were used in order to produce good codes over finite fields.

DOI
10.1109/commnet68224.2025.11288810

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Let T=Fps+u1Fps+u2Fps+u3Fps+u12Fps\mathcal{T} = {\mathbb{F}_{{p^s}}} + {u_1}{\mathbb{F}_{{p^s}}} + {u_2}{\mathbb{F}_{{p^s}}} + {u_3}{\mathbb{F}_{{p^s}}} + u_1^2{\mathbb{F}_{{p^s}}}T=Fps​+u1​Fps​+u2​Fps​+u3​Fps​+u12​Fps​ be a finite non-chain ring. In this paper, we propose a method of constructing quantum codes and LCD codes from constacyclic codes over T\mathcal{T}T. The first codes are obtained via the Gray map and the Calderbank-Shor-Steane construction from Euclidean dualcontaining constacyclic codes over T\mathcal{T}T. The second codes are obtained as the gray image of consta cyclic codes of LCD codes over T\mathcal{T}T.
Abstract In this paper, we study skew constacyclic codes over a class of non-chain rings T=Fq[u1,u2,…,ur]/⟨fi(ui),uiuj−ujui⟩i,j=1r\mathcal{T}=\mathbb{F}_q[u_1,u_2,\ldots, u_r]/ \langle f_i(u_i), u_iu_j-u_ju_i\rangle_{i,j=1}^rT=Fq​[u1​,u2​,…,ur​]/⟨fi​(ui​),ui​uj​−uj​ui​⟩i,j=1r​, where q=pmq=p^mq=pm, ppp is some odd prime, mmm is a positive integer and fi(ui)f_i(u_i)fi​(ui​) are non-constant polynomials which split into distinct linear factors. We discuss the structural properties of skew constacyclic codes over T\mathcal{T}T and their dual. We characterize (Euclidean and Hermitian) dual-containing skew constacyclic codes. These characterizations serve as a foundational framework for the development of techniques to construct quantum codes. Consequently, we derive plenty of new quantum codes including many Maximum Distance Separable (MDS) quantum codes, and many quantum codes with better parameters than existing ones. Our work further extends to the characterization of skew constacyclic Euclidean and Hermitian Linear Complementary Dual (LCD) codes over T\mathcal{T}T, and we establish that their Gray images also preserve the LCD property. From this analysis, we derive numerous Maximum Distance Separable (MDS) codes and Best-Known Linear Codes (BKLC) over Fq\mathbb{F}_qFq​. MSC Classification: MSC 11T06, MSC 81P70, MSC94B05, MSC 94B15, MSC 94B99
For an integer m ≥ 1, we study cyclic codes of length l over a commutative non-chain ring F2m + uF2m , where u2 = u . With a new Gray map and Euclidean dual-containing cyclic codes, we provide many new and superior codes to the best-known quantum error-correcting codes. Also, we characterise LCD codes of length l with respect to their generator polynomials and prove that F2m − image of an LCD code of length l is an LCD code of length 2l . Finally, we provide several optimal LCD codes from the Gray images of LCD codes over F2m + uF2m .
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In this paper, we study the structural properties of ( α + u 1 β + u 2 γ + u 1 u 2 δ ) -constacyclic codes over R = F q [ u 1 , u 2 ] / ⟨ u 1 2 − u 1 , u 2 2 − u 2 , u 1 u 2 − u 2 u 1 ⟩ where q = p m for odd prime p and m ≥ 1 . We derive the generators of constacyclic and dual constacyclic codes. We have shown that Gray image of a constacyclic code of length n is a quasi constacyclic code of length 4 n . Also we have classified all possible self dual linear codes over this ring R . We have given the applications by computing non-binary quantum codes over this ring R .