Best Known (56, ∞, s)-Nets in Base 4
(56, ∞, 66)-Net over F4 — Constructive and digital
Digital (56, m, 66)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (56, 65)-sequence over F4, using
- t-expansion [i] based on digital (49, 65)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- T6 from the second tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- t-expansion [i] based on digital (49, 65)-sequence over F4, using
(56, ∞, 91)-Net over F4 — Digital
Digital (56, m, 91)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (56, 90)-sequence over F4, using
- t-expansion [i] based on digital (50, 90)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 50 and N(F) ≥ 91, using
- t-expansion [i] based on digital (50, 90)-sequence over F4, using
(56, ∞, 182)-Net in Base 4 — Upper bound on s
There is no (56, m, 183)-net in base 4 for arbitrarily large m, because
- m-reduction [i] would yield (56, 727, 183)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4727, 183, S4, 4, 671), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 4 049832 535418 951306 730749 481876 074059 945589 013802 986985 916957 379337 000322 803802 886606 856961 343266 835497 897873 426793 038175 749021 123542 710014 968402 151805 381015 337643 140743 735376 559517 249455 888594 988700 543066 267220 585420 191957 875446 078940 369539 199424 128212 051982 458496 993558 762959 730176 819176 156632 704822 058298 090012 698629 465296 421762 317261 022024 609249 372220 275609 669270 470028 354153 694790 647734 530514 298987 530829 004890 668493 427101 628754 821120 / 7 > 4727 [i]
- extracting embedded OOA [i] would yield OOA(4727, 183, S4, 4, 671), but