Best Known (217, 217+∞, s)-Nets in Base 3
(217, 217+∞, 112)-Net over F3 — Constructive and digital
Digital (217, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (217, 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
(217, 217+∞, 163)-Net over F3 — Digital
Digital (217, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (217, 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
(217, 217+∞, 448)-Net in Base 3 — Upper bound on s
There is no (217, m, 449)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (217, 2687, 449)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32687, 449, S3, 6, 2470), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 27 602757 048330 666657 217698 833765 620573 905266 284946 365610 990337 667149 689875 196084 820181 165117 803162 767304 424252 602367 976813 676847 287192 837063 158429 953997 062872 084071 200268 713061 069845 947535 244497 140131 589026 840097 024520 409683 610648 534448 089522 782495 495516 731543 230334 604340 702369 333371 643701 252824 173716 223670 533111 160732 391577 534506 433821 549372 976957 352326 647778 515588 342071 358308 072351 779052 934152 142617 957872 822611 865365 853294 541031 515452 633307 705663 250788 494804 132727 398951 242017 260276 855418 810977 842321 068671 456972 414183 000036 217508 438231 395612 484001 516961 104652 524529 976832 630098 965389 915687 551930 017057 515391 829779 823992 219298 949762 890918 388552 008838 792552 899950 022043 206253 324172 566003 024423 904099 393730 622060 765877 095848 276047 513352 167479 949119 416479 064214 547203 258496 084503 529408 659578 891029 263364 073034 755601 823752 112015 629181 145687 930566 305722 847620 156073 810592 580212 822393 754943 992408 852393 100813 002084 405165 597952 938555 922614 393938 002125 796560 702388 072872 955276 210924 588557 075101 354676 515759 469552 490526 740304 887908 807900 822308 180076 160755 239236 863031 889363 373932 396142 710579 987453 781928 504115 575078 006914 264283 367434 265242 453026 235434 445105 293902 799096 664119 798305 966706 409487 236963 775071 053207 860075 951494 786523 713742 304877 503747 170367 935057 912149 212967 281016 386184 727909 / 2471 > 32687 [i]
- extracting embedded OOA [i] would yield OOA(32687, 449, S3, 6, 2470), but