{"title":"Stabilizer Codes Over Fields of Even Order","authors":"Simeon Ball;Edgar Moreno;Robin Simoens","doi":"10.1109/TIT.2024.3454480","DOIUrl":null,"url":null,"abstract":"We prove that the natural isomorphism <inline-formula> <tex-math>$\\mathbb {F}_{2^{h}}\\cong \\mathbb {F} _{2}^{h}$ </tex-math></inline-formula> induces a bijection between stabilizer codes on n quqits with local dimension <inline-formula> <tex-math>$q=2^{h}$ </tex-math></inline-formula> and binary stabilizer codes on hn qubits. This allows us to describe these codes geometrically: a stabilizer code over a field of even order corresponds to a so-called quantum set of symplectic polar spaces. Moreover, equivalent stabilizer codes have a similar geometry, which can be used to prove the uniqueness of a <inline-formula> <tex-math>$[\\![{4,0,3}]\\!]_{4}$ </tex-math></inline-formula> stabilizer code and the nonexistence of both a <inline-formula> <tex-math>$[\\![{7,1,4}]\\!]_{4}$ </tex-math></inline-formula> and an <inline-formula> <tex-math>$[\\![{8,0,5}]\\!]_{4}$ </tex-math></inline-formula> stabilizer code.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 5","pages":"3707-3718"},"PeriodicalIF":2.9000,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"IEEE Transactions on Information Theory","FirstCategoryId":"94","ListUrlMain":"https://ieeexplore.ieee.org/document/10664511/","RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, INFORMATION SYSTEMS","Score":null,"Total":0}
引用次数: 0
Abstract
We prove that the natural isomorphism $\mathbb {F}_{2^{h}}\cong \mathbb {F} _{2}^{h}$ induces a bijection between stabilizer codes on n quqits with local dimension $q=2^{h}$ and binary stabilizer codes on hn qubits. This allows us to describe these codes geometrically: a stabilizer code over a field of even order corresponds to a so-called quantum set of symplectic polar spaces. Moreover, equivalent stabilizer codes have a similar geometry, which can be used to prove the uniqueness of a $[\![{4,0,3}]\!]_{4}$ stabilizer code and the nonexistence of both a $[\![{7,1,4}]\!]_{4}$ and an $[\![{8,0,5}]\!]_{4}$ stabilizer code.
期刊介绍：
The IEEE Transactions on Information Theory is a journal that publishes theoretical and experimental papers concerned with the transmission, processing, and utilization of information. The boundaries of acceptable subject matter are intentionally not sharply delimited. Rather, it is hoped that as the focus of research activity changes, a flexible policy will permit this Transactions to follow suit. Current appropriate topics are best reflected by recent Tables of Contents; they are summarized in the titles of editorial areas that appear on the inside front cover.
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