Best Known (215, ∞, s)-Nets in Base 3
(215, ∞, 112)-Net over F3 — Constructive and digital
Digital (215, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (215, 111)-sequence over F3, using
- t-expansion [i] based on digital (189, 111)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on function field F4/F3 with g(F4) = 79, N(F4) ≥ 2, and N(F4) + N2(F4) ≥ 112 from GarcÃa–Stichtenoth tower as constant field extension [i]
- t-expansion [i] based on digital (189, 111)-sequence over F3, using
(215, ∞, 163)-Net over F3 — Digital
Digital (215, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (215, 162)-sequence over F3, using
- t-expansion [i] based on digital (151, 162)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 151 and N(F) ≥ 163, using
- t-expansion [i] based on digital (151, 162)-sequence over F3, using
(215, ∞, 444)-Net in Base 3 — Upper bound on s
There is no (215, m, 445)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (215, 2663, 445)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32663, 445, S3, 6, 2448), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 97 958184 640384 612858 271102 506187 745817 036141 301226 185259 758318 297945 538155 951696 944119 460963 222786 809073 295617 378516 810354 254097 463871 749435 172828 624440 574152 938300 574621 548449 344898 955227 538966 914102 454820 335029 935907 165702 326437 407247 998311 848904 099016 140775 973137 037925 627408 083286 907093 447758 546625 648621 860277 001776 540446 742319 411004 528401 646299 477223 396438 654251 916064 220929 345251 182921 154480 815026 591057 292177 991577 040912 250937 991264 938941 516440 688784 834558 640996 694285 066440 828644 932812 277746 041389 588587 687082 931919 456037 400364 494557 587703 698619 371877 918570 345060 051846 582698 798581 453015 238934 213553 152636 507191 558448 711808 531496 634845 828160 206762 099555 849119 835302 911019 207485 944037 983533 981358 275325 251196 508743 063092 462195 589555 390471 487804 550216 149833 822942 165580 016899 357704 529997 193496 046235 178912 184532 850066 116310 115499 711494 655862 356732 882349 427068 378161 097341 896226 569876 854685 705462 253499 934873 394516 969489 972905 767898 756994 347217 961732 526814 275383 198657 037199 759462 291620 186658 707819 061107 994787 219440 945529 177521 092849 944576 590359 036423 821539 280177 410311 186167 529212 919229 178209 738460 864018 935496 355768 114344 778167 682271 932171 654646 608530 882902 478370 197778 400758 545576 805328 497829 723733 194915 571240 771294 373497 222621 995563 773038 785251 795773 723767 335751 / 2449 > 32663 [i]
- extracting embedded OOA [i] would yield OOA(32663, 445, S3, 6, 2448), but