Best Known (162, 162+∞, s)-Nets in Base 3
(162, 162+∞, 85)-Net over F3 — Constructive and digital
Digital (162, m, 85)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (162, 84)-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) ≥ 85 from GarcÃa–Stichtenoth tower as constant field extension [i]
(162, 162+∞, 163)-Net over F3 — Digital
Digital (162, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (162, 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
(162, 162+∞, 338)-Net in Base 3 — Upper bound on s
There is no (162, m, 339)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (162, 1688, 339)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(31688, 339, S3, 5, 1526), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 14055 096536 363295 135553 193862 085818 740998 191623 079664 903744 302221 105956 714066 182923 802412 269050 788881 753247 108642 334528 955185 638461 459024 554799 060612 824865 930702 239317 218883 712834 402685 070275 537012 564259 864153 161707 163714 455669 498587 188719 439056 986831 284632 862652 355844 728994 066845 110740 528782 817066 568758 571044 263753 938403 576291 203345 148347 087457 105906 253653 342807 766463 466939 795787 008189 432600 826605 692116 401376 782845 252956 622073 410710 348162 161081 709842 732719 740357 556747 369079 561721 483483 708936 140033 110456 936827 207542 868451 932101 233832 351408 648154 038593 654875 535072 864142 084312 304324 512495 921851 842275 200741 715465 929067 270667 417400 435668 382100 140249 888099 526494 993383 697825 487429 127068 382928 341921 332182 400804 194630 219061 958562 537250 139914 336749 038055 616453 060240 952093 350101 925516 102285 425654 408455 814185 / 509 > 31688 [i]
- extracting embedded OOA [i] would yield OOA(31688, 339, S3, 5, 1526), but