{"title":"论平衡复正交设计的唯一性","authors":"Yiwen Gao, Yuan Li, Haibin Kan","doi":"10.1007/s10623-024-01483-x","DOIUrl":null,"url":null,"abstract":"<p><i>Complex orthogonal designs</i> (CODs) have been used to construct <i>space-time block codes</i>. Its real analog, real orthogonal designs, or equivalently, sum of squares composition formula, have a long history in mathematics. Driven by some practical considerations, Adams et al. (IEEE Trans Info Theory, 57(4):2254–2262, 2011) introduced the definition of <i>balanced complex orthogonal designs</i> (BCODs). The code rate of BCODs is 1/2, and their minimum decoding delay is proven to be <span>\\(2^m\\)</span>, where 2<i>m</i> is the number of columns. We prove, when the number of columns is fixed, all (indecomposable) balanced complex orthogonal designs (BCODs) have the same parameters <span>\\([2^m, 2m, 2^{m-1}]\\)</span>, and moreover, they are all equivalent.\n</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":null,"pages":null},"PeriodicalIF":1.4000,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the uniqueness of balanced complex orthogonal design\",\"authors\":\"Yiwen Gao, Yuan Li, Haibin Kan\",\"doi\":\"10.1007/s10623-024-01483-x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p><i>Complex orthogonal designs</i> (CODs) have been used to construct <i>space-time block codes</i>. Its real analog, real orthogonal designs, or equivalently, sum of squares composition formula, have a long history in mathematics. Driven by some practical considerations, Adams et al. (IEEE Trans Info Theory, 57(4):2254–2262, 2011) introduced the definition of <i>balanced complex orthogonal designs</i> (BCODs). The code rate of BCODs is 1/2, and their minimum decoding delay is proven to be <span>\\\\(2^m\\\\)</span>, where 2<i>m</i> is the number of columns. We prove, when the number of columns is fixed, all (indecomposable) balanced complex orthogonal designs (BCODs) have the same parameters <span>\\\\([2^m, 2m, 2^{m-1}]\\\\)</span>, and moreover, they are all equivalent.\\n</p>\",\"PeriodicalId\":11130,\"journal\":{\"name\":\"Designs, Codes and Cryptography\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.4000,\"publicationDate\":\"2024-09-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Designs, Codes and Cryptography\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10623-024-01483-x\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Designs, Codes and Cryptography","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10623-024-01483-x","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
摘要
复正交设计（COD）已被用于构建时空块编码。它的实际类似物--实正交设计，或等价于平方和组成公式，在数学中有着悠久的历史。出于一些实际考虑，亚当斯等人（IEEE Trans Info Theory, 57(4):2254-2262, 2011）提出了平衡复正交设计（BCODs）的定义。BCODs 的码率是 1/2 ，其最小解码延迟被证明为 \(2^m\)，其中 2m 是列数。我们证明，当列数固定时，所有（不可分解的）平衡复正交设计（BCODs）具有相同的参数（[2^m, 2m, 2^{m-1}]），而且，它们都是等价的。
On the uniqueness of balanced complex orthogonal design
Complex orthogonal designs (CODs) have been used to construct space-time block codes. Its real analog, real orthogonal designs, or equivalently, sum of squares composition formula, have a long history in mathematics. Driven by some practical considerations, Adams et al. (IEEE Trans Info Theory, 57(4):2254–2262, 2011) introduced the definition of balanced complex orthogonal designs (BCODs). The code rate of BCODs is 1/2, and their minimum decoding delay is proven to be \(2^m\), where 2m is the number of columns. We prove, when the number of columns is fixed, all (indecomposable) balanced complex orthogonal designs (BCODs) have the same parameters \([2^m, 2m, 2^{m-1}]\), and moreover, they are all equivalent.
期刊介绍：
Designs, Codes and Cryptography is an archival peer-reviewed technical journal publishing original research papers in the designated areas. There is a great deal of activity in design theory, coding theory and cryptography, including a substantial amount of research which brings together more than one of the subjects. While many journals exist for each of the individual areas, few encourage the interaction of the disciplines.
The journal was founded to meet the needs of mathematicians, engineers and computer scientists working in these areas, whose interests extend beyond the bounds of any one of the individual disciplines. The journal provides a forum for high quality research in its three areas, with papers touching more than one of the areas especially welcome.
The journal also considers high quality submissions in the closely related areas of finite fields and finite geometries, which provide important tools for both the construction and the actual application of designs, codes and cryptographic systems. In particular, it includes (mostly theoretical) papers on computational aspects of finite fields. It also considers topics in sequence design, which frequently admit equivalent formulations in the journal’s main areas.
Designs, Codes and Cryptography is mathematically oriented, emphasizing the algebraic and geometric aspects of the areas it covers. The journal considers high quality papers of both a theoretical and a practical nature, provided they contain a substantial amount of mathematics.
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