{"title":"On quantum codes derived from quasi-cyclic codes over a non-chain ring","authors":"Shivanshu Benjwal, Maheshanand Bhaintwal, Raj Kumar","doi":"10.1007/s11128-024-04514-7","DOIUrl":null,"url":null,"abstract":"<div><p>This paper presents a study on the structure of 1-generator quasi-cyclic (QC) codes over the non-chain ring <span>\\(R=\\mathbb {F}_{q}+u\\mathbb {F}_{q}+v\\mathbb {F}_{q}+uv\\mathbb {F}_{q}\\)</span>, where <span>\\(u^2=v^2=0,~ uv=vu\\)</span>, and <span>\\(\\mathbb {F}_q\\)</span> is a finite field of cardinality <span>\\(q=p^r\\)</span>; <i>p</i> is a prime. A minimal spanning set and size of these codes are determined. A sufficient condition for 1-generator QC codes over <i>R</i> to be free is given. BCH-type bounds on the minimum distance of free QC codes over <i>R</i> are also presented. Some optimal linear codes over <span>\\(\\mathbb {F}_q\\)</span> are obtained as the Gray images of quasi-cyclic codes over <i>R</i>. Some characterizations of the Gray images of QC codes over <i>R</i> in <span>\\(\\mathbb {F}_q\\)</span> and <span>\\(\\mathbb {F}_q+u\\mathbb {F}_q~(u^2=0)\\)</span> are done. As an application, we consider self-orthogonal subcodes of the Gray images of QC codes over <i>R</i> to obtain new and better quantum codes than those are available in the literature.</p></div>","PeriodicalId":746,"journal":{"name":"Quantum Information Processing","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Quantum Information Processing","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s11128-024-04514-7","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
This paper presents a study on the structure of 1-generator quasi-cyclic (QC) codes over the non-chain ring \(R=\mathbb {F}_{q}+u\mathbb {F}_{q}+v\mathbb {F}_{q}+uv\mathbb {F}_{q}\), where \(u^2=v^2=0,~ uv=vu\), and \(\mathbb {F}_q\) is a finite field of cardinality \(q=p^r\); p is a prime. A minimal spanning set and size of these codes are determined. A sufficient condition for 1-generator QC codes over R to be free is given. BCH-type bounds on the minimum distance of free QC codes over R are also presented. Some optimal linear codes over \(\mathbb {F}_q\) are obtained as the Gray images of quasi-cyclic codes over R. Some characterizations of the Gray images of QC codes over R in \(\mathbb {F}_q\) and \(\mathbb {F}_q+u\mathbb {F}_q~(u^2=0)\) are done. As an application, we consider self-orthogonal subcodes of the Gray images of QC codes over R to obtain new and better quantum codes than those are available in the literature.
期刊介绍:
Quantum Information Processing is a high-impact, international journal publishing cutting-edge experimental and theoretical research in all areas of Quantum Information Science. Topics of interest include quantum cryptography and communications, entanglement and discord, quantum algorithms, quantum error correction and fault tolerance, quantum computer science, quantum imaging and sensing, and experimental platforms for quantum information. Quantum Information Processing supports and inspires research by providing a comprehensive peer review process, and broadcasting high quality results in a range of formats. These include original papers, letters, broadly focused perspectives, comprehensive review articles, book reviews, and special topical issues. The journal is particularly interested in papers detailing and demonstrating quantum information protocols for cryptography, communications, computation, and sensing.