{"title":"所有极小[9,4]2-码都是双曲二次曲面","authors":"Valentino Smaldore","doi":"10.1016/j.exco.2022.100097","DOIUrl":null,"url":null,"abstract":"<div><p>Minimal codes are being intensively studied in last years. <span><math><msub><mrow><mrow><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>]</mo></mrow></mrow><mrow><mi>q</mi></mrow></msub></math></span>-minimal linear codes are in bijection with strong blocking sets of size <span><math><mi>n</mi></math></span> in <span><math><mrow><mi>P</mi><mi>G</mi><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span> and a lower bound for the size of strong blocking sets is given by <span><math><mrow><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>≤</mo><mi>n</mi></mrow></math></span>. In this note we show that all strong blocking sets of length 9 in <span><math><mrow><mi>P</mi><mi>G</mi><mrow><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> are the hyperbolic quadrics <span><math><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mo>+</mo></mrow></msup><mrow><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"3 ","pages":"Article 100097"},"PeriodicalIF":0.0000,"publicationDate":"2022-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"All minimal [9,4]2-codes are hyperbolic quadrics\",\"authors\":\"Valentino Smaldore\",\"doi\":\"10.1016/j.exco.2022.100097\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Minimal codes are being intensively studied in last years. <span><math><msub><mrow><mrow><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>]</mo></mrow></mrow><mrow><mi>q</mi></mrow></msub></math></span>-minimal linear codes are in bijection with strong blocking sets of size <span><math><mi>n</mi></math></span> in <span><math><mrow><mi>P</mi><mi>G</mi><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span> and a lower bound for the size of strong blocking sets is given by <span><math><mrow><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>≤</mo><mi>n</mi></mrow></math></span>. In this note we show that all strong blocking sets of length 9 in <span><math><mrow><mi>P</mi><mi>G</mi><mrow><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> are the hyperbolic quadrics <span><math><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mo>+</mo></mrow></msup><mrow><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>.</p></div>\",\"PeriodicalId\":100517,\"journal\":{\"name\":\"Examples and Counterexamples\",\"volume\":\"3 \",\"pages\":\"Article 100097\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-12-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Examples and Counterexamples\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S2666657X22000301\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Examples and Counterexamples","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S2666657X22000301","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Minimal codes are being intensively studied in last years. -minimal linear codes are in bijection with strong blocking sets of size in and a lower bound for the size of strong blocking sets is given by . In this note we show that all strong blocking sets of length 9 in are the hyperbolic quadrics .
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