{"title":"Bounds on Subspace Codes Based on Subspaces of Type(m,1)in Singular Linear Space","authors":"You Gao, G. Wang","doi":"10.1155/2014/497958","DOIUrl":null,"url":null,"abstract":"<jats:p>The Sphere-packing bound, Singleton bound, Wang-Xing-Safavi-Naini bound, Johnson bound, and Gilbert-Varshamov bound on the subspace codes<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M2\"><mml:mrow><mml:msub><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mi>l</mml:mi><mml:mo>,</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mi>d</mml:mi><mml:mo>,</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>based on subspaces of type<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M3\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:math>in singular linear space<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M4\"><mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant=\"double-struck\">F</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mi>l</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup></mml:mrow></mml:math>over finite fields<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M5\"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"double-struck\">F</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>are presented. Then, we prove that codes based on subspaces of type<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M6\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:math>in singular linear space attain the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures in<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M7\"><mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant=\"double-struck\">F</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mi>l</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup></mml:mrow></mml:math>.</jats:p>","PeriodicalId":49251,"journal":{"name":"Journal of Applied Mathematics","volume":"2014 1","pages":"1-9"},"PeriodicalIF":1.2000,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2014/497958","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Applied Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1155/2014/497958","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
The Sphere-packing bound, Singleton bound, Wang-Xing-Safavi-Naini bound, Johnson bound, and Gilbert-Varshamov bound on the subspace codesn+l,M,d,(m,1)qbased on subspaces of type(m,1)in singular linear spaceFq(n+l)over finite fieldsFqare presented. Then, we prove that codes based on subspaces of type(m,1)in singular linear space attain the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures inFq(n+l).
期刊介绍：
Journal of Applied Mathematics is a refereed journal devoted to the publication of original research papers and review articles in all areas of applied, computational, and industrial mathematics.
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