Best Known (232, 232+∞, s)-Nets in Base 3
(232, 232+∞, 112)-Net over F3 — Constructive and digital
Digital (232, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (232, 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
(232, 232+∞, 163)-Net over F3 — Digital
Digital (232, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (232, 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
(232, 232+∞, 479)-Net in Base 3 — Upper bound on s
There is no (232, m, 480)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (232, 2393, 480)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32393, 480, S3, 5, 2161), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 62418 045439 327992 478299 351507 310532 063980 256586 187345 001171 328004 847097 445632 866063 700752 956435 111679 968661 805296 988040 642712 458548 031569 845569 583035 025906 700813 365847 625291 848344 023905 800791 250304 025470 927044 286740 483471 540302 804644 673518 571408 421140 056940 234797 275131 963094 924149 962949 853517 022253 485865 967099 122485 262388 845411 316779 915868 835582 613052 683292 364652 264111 413816 073147 552546 048834 569275 630822 660001 344553 622119 227360 906787 772967 474524 135624 573275 093713 602740 622357 966663 766592 217927 967810 548878 721644 302017 003383 087040 182921 065680 472314 347498 311076 881668 292487 445982 000914 328492 397557 609368 812347 819456 826074 276223 431384 216204 064995 257184 758754 322669 896641 856186 335093 949574 904930 954208 611974 638438 819438 055508 555234 340864 421926 792688 389524 005893 203293 169934 281733 589699 965618 944400 045788 833189 600051 348433 559373 973523 696357 999866 052446 637805 383388 650161 935331 185572 793798 958766 604350 545206 107641 134963 978589 950650 707837 240140 761929 833879 283162 199376 041354 687110 413429 013924 583945 241453 194817 695132 821618 170785 522009 153503 392144 271019 501139 613112 107591 718338 017805 672805 755736 824628 429309 263461 133539 320504 593286 553181 131638 835161 / 1081 > 32393 [i]
- extracting embedded OOA [i] would yield OOA(32393, 480, S3, 5, 2161), but