Best Known (205, ∞, s)-Nets in Base 3
(205, ∞, 112)-Net over F3 — Constructive and digital
Digital (205, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (205, 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
(205, ∞, 163)-Net over F3 — Digital
Digital (205, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (205, 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
(205, ∞, 424)-Net in Base 3 — Upper bound on s
There is no (205, m, 425)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (205, 2543, 425)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32543, 425, S3, 6, 2338), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 55137 607210 047559 311569 107261 623016 821341 276376 913187 539073 742535 137651 872366 824841 843244 070286 625570 558661 003653 427095 281315 544595 228407 045871 033198 478670 699225 234651 373064 966305 650225 374591 082682 951512 880543 793945 624279 021459 793367 009683 320169 147742 593106 628799 641822 985578 177234 578693 684192 396889 153665 785018 809678 922561 852886 816034 831550 044476 073923 350105 188615 057037 426919 492497 518636 347395 885919 454998 816039 421233 899093 511393 251592 145787 413154 164454 596988 338730 487818 086585 082456 120772 381384 764708 796014 170248 572029 469069 773846 978163 478989 173576 879626 979083 491456 562181 268263 642552 951907 447012 556751 540425 952679 170981 896759 730710 343313 515152 689757 986807 458395 820274 085851 953987 816006 985467 308128 368687 934910 223429 151224 173584 220452 437315 622055 030898 597213 645969 532193 033557 359599 032984 060617 950375 620024 128369 555020 117538 699447 972384 073081 828177 353867 069304 673531 717858 479944 457118 361908 706332 664483 524163 060369 057088 153475 906819 642001 503427 223429 196340 391355 436650 896942 932391 935855 606243 238470 567741 613268 135685 761251 499234 741628 673488 387228 893288 395904 693145 667001 795926 723697 856727 382374 473173 823451 414201 772318 228122 561463 260418 648664 572011 264835 008509 342533 745741 261130 486626 921649 731880 296529 513872 218161 / 2339 > 32543 [i]
- extracting embedded OOA [i] would yield OOA(32543, 425, S3, 6, 2338), but