Best Known (229, 229+∞, s)-Nets in Base 3
(229, 229+∞, 112)-Net over F3 — Constructive and digital
Digital (229, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (229, 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
(229, 229+∞, 163)-Net over F3 — Digital
Digital (229, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (229, 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
(229, 229+∞, 473)-Net in Base 3 — Upper bound on s
There is no (229, m, 474)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (229, 2363, 474)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32363, 474, S3, 5, 2134), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 613 715029 614520 758681 704113 021330 535129 458472 571634 619789 102175 365602 283010 586316 643579 045636 436879 558084 792763 271461 419487 748035 774235 771883 140233 194730 584900 773760 355094 522574 539477 568743 395508 162427 480554 470687 340692 619678 373181 357122 595859 249703 018393 958768 801798 779600 608461 385357 018716 339327 447484 572210 428026 048754 000497 712960 598384 932849 190786 324567 459336 328278 511447 021348 845299 188236 453183 270861 182385 024610 829892 600582 077139 600797 887757 353734 012111 860139 805748 196349 101589 423616 927266 110650 258051 928368 466822 141008 746266 903785 121810 625262 789713 317582 057071 963203 603846 807703 844908 591053 222444 477559 918860 573558 466019 568080 923443 971096 189753 792814 983181 454937 993017 662569 865787 151155 981626 115211 880087 439200 467713 372198 937281 581138 089919 292482 050703 906434 210979 850906 654080 881565 281842 614604 085080 136791 129815 413575 752984 645896 058002 474952 461534 383063 969597 518469 424869 666932 965368 852227 997496 888756 780475 761177 329224 899652 793776 643904 118176 497701 083394 633129 600965 617428 996356 979162 116440 097379 793450 587237 889377 394974 636616 830886 252592 655395 490409 980037 111310 612640 094465 030077 727821 836081 583663 703725 959119 896402 134907 / 2135 > 32363 [i]
- extracting embedded OOA [i] would yield OOA(32363, 474, S3, 5, 2134), but