Quasi-symmetric 2-(41, 9, 9) designs and doubly even self-dual codes of length 40

Abstract The existence of a quasi-symmetric 2-(41, 9, 9) design with intersection numbers %$x=1, y=3%$ is a long-standing open question. Using linear codes and properties of subdesigns, we prove that a cyclic quasi-symmetric 2-(41, 9, 9) design does not exist, and if %$p<41%$ is a prime number be...
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Gespeichert in:

Autor*in:	Munemasa, Akihiro [verfasserIn] Tonchev, Vladimir D.

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2022

Schlagwörter:	Quasi-symmetric design Subdesign Cyclic code Self-dual code Automorphism group

Anmerkung:	© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022

Übergeordnetes Werk:	Enthalten in: Applicable algebra in engineering, communication and computing - Berlin : Springer, 1990, 33(2022), 6 vom: 10. Feb., Seite 855-866
Übergeordnetes Werk:	volume:33 ; year:2022 ; number:6 ; day:10 ; month:02 ; pages:855-866

Links:	Volltext

DOI / URN:	10.1007/s00200-022-00543-w

Katalog-ID:	SPR048649406

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520			\|a Abstract The existence of a quasi-symmetric 2-(41, 9, 9) design with intersection numbers %$x=1, y=3%$ is a long-standing open question. Using linear codes and properties of subdesigns, we prove that a cyclic quasi-symmetric 2-(41, 9, 9) design does not exist, and if %$p<41%$ is a prime number being the order of an automorphism of a quasi-symmetric 2-(41, 9, 9) design, then %$p\le 5%$. The derived design with respect to a point of a quasi-symmetric 2-(41, 9, 9) design with block intersection numbers 1 and 3 is a quasi-symmetric 1-(40, 8, 9) design with block intersection numbers 0 and 2. The incidence matrix of the latter generates a binary doubly even code of length 40. Using the database of binary doubly even self-dual codes of length 40 classified by Betsumiya et al. (Electron J Combin 19(P18):12, 2012), we prove that there is no quasi-symmetric 2-(41, 9, 9) design with an automorphism %$\phi %$ of order 5 with exactly one fixed point such that the binary code of the derived design is contained in a doubly-even self-dual [40, 20] code invariant under %$\phi %$.
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allfields	10.1007/s00200-022-00543-w doi (DE-627)SPR048649406 (SPR)s00200-022-00543-w-e DE-627 ger DE-627 rakwb eng Munemasa, Akihiro verfasserin aut Quasi-symmetric 2-(41, 9, 9) designs and doubly even self-dual codes of length 40 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022 Abstract The existence of a quasi-symmetric 2-(41, 9, 9) design with intersection numbers %$x=1, y=3%$ is a long-standing open question. Using linear codes and properties of subdesigns, we prove that a cyclic quasi-symmetric 2-(41, 9, 9) design does not exist, and if %$p<41%$ is a prime number being the order of an automorphism of a quasi-symmetric 2-(41, 9, 9) design, then %$p\le 5%$. The derived design with respect to a point of a quasi-symmetric 2-(41, 9, 9) design with block intersection numbers 1 and 3 is a quasi-symmetric 1-(40, 8, 9) design with block intersection numbers 0 and 2. The incidence matrix of the latter generates a binary doubly even code of length 40. Using the database of binary doubly even self-dual codes of length 40 classified by Betsumiya et al. (Electron J Combin 19(P18):12, 2012), we prove that there is no quasi-symmetric 2-(41, 9, 9) design with an automorphism %$\phi %$ of order 5 with exactly one fixed point such that the binary code of the derived design is contained in a doubly-even self-dual [40, 20] code invariant under %$\phi %$. Quasi-symmetric design (dpeaa)DE-He213 Subdesign (dpeaa)DE-He213 Cyclic code (dpeaa)DE-He213 Self-dual code (dpeaa)DE-He213 Automorphism group (dpeaa)DE-He213 Tonchev, Vladimir D. aut Enthalten in Applicable algebra in engineering, communication and computing Berlin : Springer, 1990 33(2022), 6 vom: 10. Feb., Seite 855-866 (DE-627)253389909 (DE-600)1458434-7 1432-0622 nnns volume:33 year:2022 number:6 day:10 month:02 pages:855-866 https://dx.doi.org/10.1007/s00200-022-00543-w lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 33 2022 6 10 02 855-866
spelling	10.1007/s00200-022-00543-w doi (DE-627)SPR048649406 (SPR)s00200-022-00543-w-e DE-627 ger DE-627 rakwb eng Munemasa, Akihiro verfasserin aut Quasi-symmetric 2-(41, 9, 9) designs and doubly even self-dual codes of length 40 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022 Abstract The existence of a quasi-symmetric 2-(41, 9, 9) design with intersection numbers %$x=1, y=3%$ is a long-standing open question. Using linear codes and properties of subdesigns, we prove that a cyclic quasi-symmetric 2-(41, 9, 9) design does not exist, and if %$p<41%$ is a prime number being the order of an automorphism of a quasi-symmetric 2-(41, 9, 9) design, then %$p\le 5%$. The derived design with respect to a point of a quasi-symmetric 2-(41, 9, 9) design with block intersection numbers 1 and 3 is a quasi-symmetric 1-(40, 8, 9) design with block intersection numbers 0 and 2. The incidence matrix of the latter generates a binary doubly even code of length 40. Using the database of binary doubly even self-dual codes of length 40 classified by Betsumiya et al. (Electron J Combin 19(P18):12, 2012), we prove that there is no quasi-symmetric 2-(41, 9, 9) design with an automorphism %$\phi %$ of order 5 with exactly one fixed point such that the binary code of the derived design is contained in a doubly-even self-dual [40, 20] code invariant under %$\phi %$. Quasi-symmetric design (dpeaa)DE-He213 Subdesign (dpeaa)DE-He213 Cyclic code (dpeaa)DE-He213 Self-dual code (dpeaa)DE-He213 Automorphism group (dpeaa)DE-He213 Tonchev, Vladimir D. aut Enthalten in Applicable algebra in engineering, communication and computing Berlin : Springer, 1990 33(2022), 6 vom: 10. Feb., Seite 855-866 (DE-627)253389909 (DE-600)1458434-7 1432-0622 nnns volume:33 year:2022 number:6 day:10 month:02 pages:855-866 https://dx.doi.org/10.1007/s00200-022-00543-w lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 33 2022 6 10 02 855-866
allfields_unstemmed	10.1007/s00200-022-00543-w doi (DE-627)SPR048649406 (SPR)s00200-022-00543-w-e DE-627 ger DE-627 rakwb eng Munemasa, Akihiro verfasserin aut Quasi-symmetric 2-(41, 9, 9) designs and doubly even self-dual codes of length 40 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022 Abstract The existence of a quasi-symmetric 2-(41, 9, 9) design with intersection numbers %$x=1, y=3%$ is a long-standing open question. Using linear codes and properties of subdesigns, we prove that a cyclic quasi-symmetric 2-(41, 9, 9) design does not exist, and if %$p<41%$ is a prime number being the order of an automorphism of a quasi-symmetric 2-(41, 9, 9) design, then %$p\le 5%$. The derived design with respect to a point of a quasi-symmetric 2-(41, 9, 9) design with block intersection numbers 1 and 3 is a quasi-symmetric 1-(40, 8, 9) design with block intersection numbers 0 and 2. The incidence matrix of the latter generates a binary doubly even code of length 40. Using the database of binary doubly even self-dual codes of length 40 classified by Betsumiya et al. (Electron J Combin 19(P18):12, 2012), we prove that there is no quasi-symmetric 2-(41, 9, 9) design with an automorphism %$\phi %$ of order 5 with exactly one fixed point such that the binary code of the derived design is contained in a doubly-even self-dual [40, 20] code invariant under %$\phi %$. Quasi-symmetric design (dpeaa)DE-He213 Subdesign (dpeaa)DE-He213 Cyclic code (dpeaa)DE-He213 Self-dual code (dpeaa)DE-He213 Automorphism group (dpeaa)DE-He213 Tonchev, Vladimir D. aut Enthalten in Applicable algebra in engineering, communication and computing Berlin : Springer, 1990 33(2022), 6 vom: 10. Feb., Seite 855-866 (DE-627)253389909 (DE-600)1458434-7 1432-0622 nnns volume:33 year:2022 number:6 day:10 month:02 pages:855-866 https://dx.doi.org/10.1007/s00200-022-00543-w lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 33 2022 6 10 02 855-866
allfieldsGer	10.1007/s00200-022-00543-w doi (DE-627)SPR048649406 (SPR)s00200-022-00543-w-e DE-627 ger DE-627 rakwb eng Munemasa, Akihiro verfasserin aut Quasi-symmetric 2-(41, 9, 9) designs and doubly even self-dual codes of length 40 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022 Abstract The existence of a quasi-symmetric 2-(41, 9, 9) design with intersection numbers %$x=1, y=3%$ is a long-standing open question. Using linear codes and properties of subdesigns, we prove that a cyclic quasi-symmetric 2-(41, 9, 9) design does not exist, and if %$p<41%$ is a prime number being the order of an automorphism of a quasi-symmetric 2-(41, 9, 9) design, then %$p\le 5%$. The derived design with respect to a point of a quasi-symmetric 2-(41, 9, 9) design with block intersection numbers 1 and 3 is a quasi-symmetric 1-(40, 8, 9) design with block intersection numbers 0 and 2. The incidence matrix of the latter generates a binary doubly even code of length 40. Using the database of binary doubly even self-dual codes of length 40 classified by Betsumiya et al. (Electron J Combin 19(P18):12, 2012), we prove that there is no quasi-symmetric 2-(41, 9, 9) design with an automorphism %$\phi %$ of order 5 with exactly one fixed point such that the binary code of the derived design is contained in a doubly-even self-dual [40, 20] code invariant under %$\phi %$. Quasi-symmetric design (dpeaa)DE-He213 Subdesign (dpeaa)DE-He213 Cyclic code (dpeaa)DE-He213 Self-dual code (dpeaa)DE-He213 Automorphism group (dpeaa)DE-He213 Tonchev, Vladimir D. aut Enthalten in Applicable algebra in engineering, communication and computing Berlin : Springer, 1990 33(2022), 6 vom: 10. Feb., Seite 855-866 (DE-627)253389909 (DE-600)1458434-7 1432-0622 nnns volume:33 year:2022 number:6 day:10 month:02 pages:855-866 https://dx.doi.org/10.1007/s00200-022-00543-w lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 33 2022 6 10 02 855-866
allfieldsSound	10.1007/s00200-022-00543-w doi (DE-627)SPR048649406 (SPR)s00200-022-00543-w-e DE-627 ger DE-627 rakwb eng Munemasa, Akihiro verfasserin aut Quasi-symmetric 2-(41, 9, 9) designs and doubly even self-dual codes of length 40 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022 Abstract The existence of a quasi-symmetric 2-(41, 9, 9) design with intersection numbers %$x=1, y=3%$ is a long-standing open question. Using linear codes and properties of subdesigns, we prove that a cyclic quasi-symmetric 2-(41, 9, 9) design does not exist, and if %$p<41%$ is a prime number being the order of an automorphism of a quasi-symmetric 2-(41, 9, 9) design, then %$p\le 5%$. The derived design with respect to a point of a quasi-symmetric 2-(41, 9, 9) design with block intersection numbers 1 and 3 is a quasi-symmetric 1-(40, 8, 9) design with block intersection numbers 0 and 2. The incidence matrix of the latter generates a binary doubly even code of length 40. Using the database of binary doubly even self-dual codes of length 40 classified by Betsumiya et al. (Electron J Combin 19(P18):12, 2012), we prove that there is no quasi-symmetric 2-(41, 9, 9) design with an automorphism %$\phi %$ of order 5 with exactly one fixed point such that the binary code of the derived design is contained in a doubly-even self-dual [40, 20] code invariant under %$\phi %$. Quasi-symmetric design (dpeaa)DE-He213 Subdesign (dpeaa)DE-He213 Cyclic code (dpeaa)DE-He213 Self-dual code (dpeaa)DE-He213 Automorphism group (dpeaa)DE-He213 Tonchev, Vladimir D. aut Enthalten in Applicable algebra in engineering, communication and computing Berlin : Springer, 1990 33(2022), 6 vom: 10. Feb., Seite 855-866 (DE-627)253389909 (DE-600)1458434-7 1432-0622 nnns volume:33 year:2022 number:6 day:10 month:02 pages:855-866 https://dx.doi.org/10.1007/s00200-022-00543-w lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 33 2022 6 10 02 855-866
language	English
source	Enthalten in Applicable algebra in engineering, communication and computing 33(2022), 6 vom: 10. Feb., Seite 855-866 volume:33 year:2022 number:6 day:10 month:02 pages:855-866
sourceStr	Enthalten in Applicable algebra in engineering, communication and computing 33(2022), 6 vom: 10. Feb., Seite 855-866 volume:33 year:2022 number:6 day:10 month:02 pages:855-866
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Quasi-symmetric design Subdesign Cyclic code Self-dual code Automorphism group
isfreeaccess_bool	false
container_title	Applicable algebra in engineering, communication and computing
authorswithroles_txt_mv	Munemasa, Akihiro @@aut@@ Tonchev, Vladimir D. @@aut@@
publishDateDaySort_date	2022-02-10T00:00:00Z
hierarchy_top_id	253389909
id	SPR048649406
language_de	englisch
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author	Munemasa, Akihiro
spellingShingle	Munemasa, Akihiro misc Quasi-symmetric design misc Subdesign misc Cyclic code misc Self-dual code misc Automorphism group Quasi-symmetric 2-(41, 9, 9) designs and doubly even self-dual codes of length 40
authorStr	Munemasa, Akihiro
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topic_title	Quasi-symmetric 2-(41, 9, 9) designs and doubly even self-dual codes of length 40 Quasi-symmetric design (dpeaa)DE-He213 Subdesign (dpeaa)DE-He213 Cyclic code (dpeaa)DE-He213 Self-dual code (dpeaa)DE-He213 Automorphism group (dpeaa)DE-He213
topic	misc Quasi-symmetric design misc Subdesign misc Cyclic code misc Self-dual code misc Automorphism group
topic_unstemmed	misc Quasi-symmetric design misc Subdesign misc Cyclic code misc Self-dual code misc Automorphism group
topic_browse	misc Quasi-symmetric design misc Subdesign misc Cyclic code misc Self-dual code misc Automorphism group
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title	Quasi-symmetric 2-(41, 9, 9) designs and doubly even self-dual codes of length 40
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title_full	Quasi-symmetric 2-(41, 9, 9) designs and doubly even self-dual codes of length 40
author_sort	Munemasa, Akihiro
journal	Applicable algebra in engineering, communication and computing
journalStr	Applicable algebra in engineering, communication and computing
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author_browse	Munemasa, Akihiro Tonchev, Vladimir D.
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author-letter	Munemasa, Akihiro
doi_str_mv	10.1007/s00200-022-00543-w
title_sort	quasi-symmetric 2-(41, 9, 9) designs and doubly even self-dual codes of length 40
title_auth	Quasi-symmetric 2-(41, 9, 9) designs and doubly even self-dual codes of length 40
abstract	Abstract The existence of a quasi-symmetric 2-(41, 9, 9) design with intersection numbers %$x=1, y=3%$ is a long-standing open question. Using linear codes and properties of subdesigns, we prove that a cyclic quasi-symmetric 2-(41, 9, 9) design does not exist, and if %$p<41%$ is a prime number being the order of an automorphism of a quasi-symmetric 2-(41, 9, 9) design, then %$p\le 5%$. The derived design with respect to a point of a quasi-symmetric 2-(41, 9, 9) design with block intersection numbers 1 and 3 is a quasi-symmetric 1-(40, 8, 9) design with block intersection numbers 0 and 2. The incidence matrix of the latter generates a binary doubly even code of length 40. Using the database of binary doubly even self-dual codes of length 40 classified by Betsumiya et al. (Electron J Combin 19(P18):12, 2012), we prove that there is no quasi-symmetric 2-(41, 9, 9) design with an automorphism %$\phi %$ of order 5 with exactly one fixed point such that the binary code of the derived design is contained in a doubly-even self-dual [40, 20] code invariant under %$\phi %$. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022
abstractGer	Abstract The existence of a quasi-symmetric 2-(41, 9, 9) design with intersection numbers %$x=1, y=3%$ is a long-standing open question. Using linear codes and properties of subdesigns, we prove that a cyclic quasi-symmetric 2-(41, 9, 9) design does not exist, and if %$p<41%$ is a prime number being the order of an automorphism of a quasi-symmetric 2-(41, 9, 9) design, then %$p\le 5%$. The derived design with respect to a point of a quasi-symmetric 2-(41, 9, 9) design with block intersection numbers 1 and 3 is a quasi-symmetric 1-(40, 8, 9) design with block intersection numbers 0 and 2. The incidence matrix of the latter generates a binary doubly even code of length 40. Using the database of binary doubly even self-dual codes of length 40 classified by Betsumiya et al. (Electron J Combin 19(P18):12, 2012), we prove that there is no quasi-symmetric 2-(41, 9, 9) design with an automorphism %$\phi %$ of order 5 with exactly one fixed point such that the binary code of the derived design is contained in a doubly-even self-dual [40, 20] code invariant under %$\phi %$. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022
abstract_unstemmed	Abstract The existence of a quasi-symmetric 2-(41, 9, 9) design with intersection numbers %$x=1, y=3%$ is a long-standing open question. Using linear codes and properties of subdesigns, we prove that a cyclic quasi-symmetric 2-(41, 9, 9) design does not exist, and if %$p<41%$ is a prime number being the order of an automorphism of a quasi-symmetric 2-(41, 9, 9) design, then %$p\le 5%$. The derived design with respect to a point of a quasi-symmetric 2-(41, 9, 9) design with block intersection numbers 1 and 3 is a quasi-symmetric 1-(40, 8, 9) design with block intersection numbers 0 and 2. The incidence matrix of the latter generates a binary doubly even code of length 40. Using the database of binary doubly even self-dual codes of length 40 classified by Betsumiya et al. (Electron J Combin 19(P18):12, 2012), we prove that there is no quasi-symmetric 2-(41, 9, 9) design with an automorphism %$\phi %$ of order 5 with exactly one fixed point such that the binary code of the derived design is contained in a doubly-even self-dual [40, 20] code invariant under %$\phi %$. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022
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title_short	Quasi-symmetric 2-(41, 9, 9) designs and doubly even self-dual codes of length 40
url	https://dx.doi.org/10.1007/s00200-022-00543-w
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author2	Tonchev, Vladimir D.
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doi_str	10.1007/s00200-022-00543-w
up_date	2024-07-03T20:35:05.684Z
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fullrecord_marcxml	<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000caa a22002652 4500</leader><controlfield tag="001">SPR048649406</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20230509120324.0</controlfield><controlfield tag="007">cr uuu---uuuuu</controlfield><controlfield tag="008">221118s2022 xx \|\|\|\|\|o 00\| \|\|eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/s00200-022-00543-w</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)SPR048649406</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(SPR)s00200-022-00543-w-e</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Munemasa, Akihiro</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Quasi-symmetric 2-(41, 9, 9) designs and doubly even self-dual codes of length 40</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2022</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">Computermedien</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Online-Ressource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract The existence of a quasi-symmetric 2-(41, 9, 9) design with intersection numbers %$x=1, y=3%$ is a long-standing open question. Using linear codes and properties of subdesigns, we prove that a cyclic quasi-symmetric 2-(41, 9, 9) design does not exist, and if %$p<41%$ is a prime number being the order of an automorphism of a quasi-symmetric 2-(41, 9, 9) design, then %$p\le 5%$. The derived design with respect to a point of a quasi-symmetric 2-(41, 9, 9) design with block intersection numbers 1 and 3 is a quasi-symmetric 1-(40, 8, 9) design with block intersection numbers 0 and 2. The incidence matrix of the latter generates a binary doubly even code of length 40. Using the database of binary doubly even self-dual codes of length 40 classified by Betsumiya et al. (Electron J Combin 19(P18):12, 2012), we prove that there is no quasi-symmetric 2-(41, 9, 9) design with an automorphism %$\phi %$ of order 5 with exactly one fixed point such that the binary code of the derived design is contained in a doubly-even self-dual [40, 20] code invariant under %$\phi %$.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Quasi-symmetric design</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Subdesign</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Cyclic code</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Self-dual code</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Automorphism group</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Tonchev, Vladimir D.</subfield><subfield code="4">aut</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Applicable algebra in engineering, communication and computing</subfield><subfield code="d">Berlin : Springer, 1990</subfield><subfield code="g">33(2022), 6 vom: 10. 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Quasi-symmetric 2-(41, 9, 9) designs and doubly even self-dual codes of length 40

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