Best Known (58, 58+∞, s)-Nets in Base 5
(58, 58+∞, 82)-Net over F5 — Constructive and digital
Digital (58, m, 82)-net over F5 for arbitrarily large m, using
- net from sequence [i] based on digital (58, 81)-sequence over F5, using
- t-expansion [i] based on digital (48, 81)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 48 and N(F) ≥ 82, using
- t-expansion [i] based on digital (48, 81)-sequence over F5, using
(58, 58+∞, 112)-Net over F5 — Digital
Digital (58, m, 112)-net over F5 for arbitrarily large m, using
- net from sequence [i] based on digital (58, 111)-sequence over F5, using
- t-expansion [i] based on digital (57, 111)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 57 and N(F) ≥ 112, using
- t-expansion [i] based on digital (57, 111)-sequence over F5, using
(58, 58+∞, 249)-Net in Base 5 — Upper bound on s
There is no (58, m, 250)-net in base 5 for arbitrarily large m, because
- m-reduction [i] would yield (58, 995, 250)-net in base 5, but
- extracting embedded OOA [i] would yield OOA(5995, 250, S5, 4, 937), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 186 652723 700643 775798 017908 944763 433923 418289 274341 604924 342867 959193 382195 155126 890888 065419 576220 471918 997986 064848 524843 097504 270806 478968 304163 440786 151246 882133 227665 030054 799015 197180 366302 220098 159253 022623 648102 502959 186758 161035 654225 083020 762139 675770 885296 223893 962845 732191 844403 532582 088559 691233 889777 429493 305601 265673 690529 485852 365972 433040 558639 057898 721423 570132 733748 213087 961106 143627 264119 968965 208390 820242 645925 973900 438902 921980 842921 733672 248958 590406 965372 923531 585383 209484 013187 277808 347579 164423 673015 609111 325688 854785 077503 425570 959356 311269 280742 975536 353379 971078 413853 087884 801742 394734 940349 972525 338149 459352 507160 785875 246766 796209 385574 911721 050739 288330 078125 / 469 > 5995 [i]
- extracting embedded OOA [i] would yield OOA(5995, 250, S5, 4, 937), but