Best Known (216, 216+∞, s)-Nets in Base 3
(216, 216+∞, 112)-Net over F3 — Constructive and digital
Digital (216, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (216, 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
(216, 216+∞, 163)-Net over F3 — Digital
Digital (216, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (216, 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
(216, 216+∞, 446)-Net in Base 3 — Upper bound on s
There is no (216, m, 447)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (216, 2675, 447)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32675, 447, S3, 6, 2459), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 1 733307 545924 969612 191000 156857 332643 992874 414590 910916 760396 547190 328091 332923 820315 400600 641706 487133 663510 435892 576594 871158 900687 286481 464698 981966 748991 830010 124540 351388 696418 795550 272419 686978 333132 116000 448542 921654 870599 153205 755706 836477 049202 378724 722840 709180 935438 927665 341261 293708 538741 149435 761482 636777 190202 658046 342288 181480 792973 616732 235696 642813 747971 558534 342655 475755 661580 361338 390184 574367 346218 862378 626841 528557 358303 173977 869031 649295 772311 329279 011423 463871 482044 112515 765080 920328 484777 635100 165351 829151 005870 508828 976346 316131 681991 693845 835489 539142 278914 226008 248307 966342 321617 357798 545673 240773 495862 027543 575144 816721 944720 495683 525530 887733 310907 830757 725844 100109 097213 222362 586211 578350 855011 307580 893549 666108 564639 691395 673093 281303 240279 069271 156992 722062 862499 323561 161696 382719 618393 365050 111576 482790 790189 448808 416985 360198 382365 375161 313193 037704 247708 266817 916771 277265 223066 244431 732964 108629 344607 940256 293659 814528 580656 541148 895342 920312 177709 442732 399563 690610 647216 900746 792396 439295 123713 520212 345469 124284 887533 072480 232405 916967 716242 519101 276563 609668 217595 722669 767062 238340 266504 930033 584217 215155 273900 432496 232551 146935 990980 914588 997570 882502 990002 680432 580063 589140 328802 682230 066367 528871 546484 886018 882277 481709 / 82 > 32675 [i]
- extracting embedded OOA [i] would yield OOA(32675, 447, S3, 6, 2459), but