Best Known (196, ∞, s)-Nets in Base 3
(196, ∞, 112)-Net over F3 — Constructive and digital
Digital (196, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (196, 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
(196, ∞, 163)-Net over F3 — Digital
Digital (196, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (196, 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
(196, ∞, 406)-Net in Base 3 — Upper bound on s
There is no (196, m, 407)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (196, 2435, 407)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32435, 407, S3, 6, 2239), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 1 647276 503421 979953 520590 491289 388642 958006 323522 000529 233451 368992 198248 673569 827535 431224 187633 152819 811880 482216 104564 929661 687626 625421 798235 952693 674066 550954 759094 686098 254825 842663 987817 769012 907901 162342 291627 439228 964259 810721 430169 640340 137076 039727 598876 882957 194920 737393 523617 175506 702399 750588 300898 661601 298076 875465 230051 042001 549169 945778 720350 438156 223181 645211 902580 465871 693109 451293 897265 568938 279158 458922 757792 770623 009000 450783 824762 160668 839350 065613 653672 033442 412497 839259 655524 109537 643051 640802 647021 595436 987620 361055 241420 650560 173347 971171 033687 319090 565787 088442 987945 957752 580893 270039 994809 063838 874581 896531 190850 514647 750217 847651 385447 507343 857966 671557 529308 750609 798547 365142 221458 299120 781250 792126 447222 574455 381746 140653 600451 559770 956968 414744 145563 092514 119091 347558 712577 199447 550447 969786 010918 369067 936520 710530 610792 681847 433654 654607 962448 439096 484228 700458 756843 713344 443373 461446 105025 428926 841373 230316 075424 609920 856729 866598 530242 336307 679261 369947 788106 816094 431687 230335 285669 775834 351049 367232 959157 857877 458288 687980 435049 876108 522178 324539 863931 386331 647676 959057 793666 352406 714946 309972 574485 598242 953769 / 224 > 32435 [i]
- extracting embedded OOA [i] would yield OOA(32435, 407, S3, 6, 2239), but