{"title":"On the codewords of generalized Reed-Muller codes reaching the fourth weight","authors":"Somayyeh Golalizadeh, Nasrin Soltankhah","doi":"10.1016/j.ic.2023.105113","DOIUrl":null,"url":null,"abstract":"<div><p>Recently, the fourth weight of generalized Reed-Muller codes <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>a</mi><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mi>b</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> have been characterized for <span><math><mn>3</mn><mo>≤</mo><mi>b</mi><mo><</mo><mfrac><mrow><mi>q</mi><mo>+</mo><mn>4</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math></span> and <span><math><mn>0</mn><mo>≤</mo><mi>a</mi><mo>≤</mo><mi>m</mi><mo>−</mo><mn>3</mn></math></span>. In this paper, we study the codewords reaching the fourth weight of generalized Reed-Muller codes. We will determine the fourth weight codewords of generalized Reed-Muller codes of order <span><math><mi>r</mi><mo>=</mo><mi>a</mi><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mi>b</mi></math></span> with <span><math><mn>3</mn><mo>≤</mo><mi>b</mi><mo><</mo><mfrac><mrow><mi>q</mi><mo>+</mo><mn>4</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math></span> and <span><math><mn>0</mn><mo>≤</mo><mi>a</mi><mo>≤</mo><mi>m</mi><mo>−</mo><mn>3</mn></math></span>. Also, we characterize the number of fourth weight codewords of <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>b</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> for <span><math><mn>3</mn><mo>≤</mo><mi>b</mi><mo><</mo><mfrac><mrow><mi>q</mi><mo>+</mo><mn>4</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math></span>.</p></div>","PeriodicalId":54985,"journal":{"name":"Information and Computation","volume":"296 ","pages":"Article 105113"},"PeriodicalIF":0.8000,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Information and Computation","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0890540123001165","RegionNum":4,"RegionCategory":"计算机科学","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
Abstract
Recently, the fourth weight of generalized Reed-Muller codes have been characterized for and . In this paper, we study the codewords reaching the fourth weight of generalized Reed-Muller codes. We will determine the fourth weight codewords of generalized Reed-Muller codes of order with and . Also, we characterize the number of fourth weight codewords of for .
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Information and Computation welcomes original papers in all areas of theoretical computer science and computational applications of information theory. Survey articles of exceptional quality will also be considered. Particularly welcome are papers contributing new results in active theoretical areas such as
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