Best Known (220, ∞, s)-Nets in Base 3
(220, ∞, 112)-Net over F3 — Constructive and digital
Digital (220, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (220, 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
(220, ∞, 163)-Net over F3 — Digital
Digital (220, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (220, 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
(220, ∞, 454)-Net in Base 3 — Upper bound on s
There is no (220, m, 455)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (220, 2723, 455)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32723, 455, S3, 6, 2503), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 2 064361 508368 531451 448610 492656 157314 907184 791566 546706 540368 445812 412275 429153 465423 126347 871183 052080 423944 073841 753548 266122 287233 727336 121139 244199 431503 142380 627907 961111 027810 487128 002805 813634 383818 212468 456349 284918 983014 377649 441739 930248 702257 608643 365579 279150 710491 183791 675840 244919 546083 174048 459922 594539 813831 672138 020422 266351 094976 925692 573617 067503 774481 333076 710776 345187 598902 015450 129428 153904 077738 400852 947648 866228 604299 465484 785246 071467 410593 386186 614753 509714 507909 968022 596056 046457 822923 682816 216735 004631 299177 362130 306326 133624 015411 378497 886882 260923 821370 766827 667562 820805 044188 476585 213485 319974 160603 244653 646618 333640 187945 791265 136124 776813 368933 853816 668853 856686 524802 474691 708450 524732 243398 464233 698772 096271 369654 157163 051476 239287 009569 626095 248767 111616 916196 540860 410410 296107 582387 285772 999041 145207 131508 141525 422041 416652 852127 593510 664872 838349 305124 623491 653232 804160 002510 727947 903314 985035 552551 775270 040788 895589 630985 390894 740130 342951 456846 500441 695090 118244 990207 563296 114189 029462 242839 169965 222501 143284 575999 456483 472320 339457 548055 831460 027425 728341 232019 098930 707225 355141 310628 804520 645476 220388 113125 609086 187979 872285 841838 649462 342429 893776 937231 295907 724640 050147 017621 402730 913129 983663 990210 154278 917907 793317 969139 540509 590689 958273 / 1252 > 32723 [i]
- extracting embedded OOA [i] would yield OOA(32723, 455, S3, 6, 2503), but