Best Known (165, ∞, s)-Nets in Base 3
(165, ∞, 88)-Net over F3 — Constructive and digital
Digital (165, m, 88)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (165, 87)-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) ≥ 88 from GarcÃa–Stichtenoth tower as constant field extension [i]
(165, ∞, 163)-Net over F3 — Digital
Digital (165, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (165, 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
(165, ∞, 344)-Net in Base 3 — Upper bound on s
There is no (165, m, 345)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (165, 1718, 345)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(31718, 345, S3, 5, 1553), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 1 424649 716958 236962 845238 179330 597094 194950 469536 259629 363915 139115 379288 973560 330894 965441 949799 388165 207782 765863 544553 924070 992764 013504 956066 381389 933121 065244 722920 437813 438664 970785 980017 037115 698702 215313 386904 232507 580994 605047 802345 706669 852253 337019 515152 385553 354658 457490 746321 551881 485929 120448 276023 294225 583207 700657 318629 989413 011885 508904 949259 228197 972085 295836 126089 024860 058171 084597 121539 326712 764948 294490 971538 613945 665709 188146 601550 771394 799149 604517 001945 276374 702580 929531 390372 788272 125281 421791 449827 501718 783636 174488 300924 649136 453878 196232 869028 952328 843342 989994 095380 956964 377234 974708 699430 989088 124156 847973 366666 825898 151315 372493 249075 830283 972357 034555 511125 508156 023442 237155 975294 936759 228151 080634 810933 841123 149410 909627 728696 854132 365800 587512 015223 260875 426959 123021 411336 268832 / 259 > 31718 [i]
- extracting embedded OOA [i] would yield OOA(31718, 345, S3, 5, 1553), but