Best Known (34, ∞, s)-Nets in Base 4
(34, ∞, 56)-Net over F4 — Constructive and digital
Digital (34, m, 56)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (34, 55)-sequence over F4, using
- t-expansion [i] based on digital (33, 55)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 33 and N(F) ≥ 56, using
- F5 from the 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) = 33 and N(F) ≥ 56, using
- t-expansion [i] based on digital (33, 55)-sequence over F4, using
(34, ∞, 65)-Net over F4 — Digital
Digital (34, m, 65)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (34, 64)-sequence over F4, using
- t-expansion [i] based on digital (33, 64)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 33 and N(F) ≥ 65, using
- t-expansion [i] based on digital (33, 64)-sequence over F4, using
(34, ∞, 115)-Net in Base 4 — Upper bound on s
There is no (34, m, 116)-net in base 4 for arbitrarily large m, because
- m-reduction [i] would yield (34, 459, 116)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4459, 116, S4, 4, 425), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 194 992852 130599 185118 421166 892634 703653 381612 518469 560737 770682 006780 472144 656754 839382 785075 398920 255753 826638 366254 610478 883239 042686 972447 623448 278025 876088 293543 397451 145808 333760 597434 939212 472810 192178 080192 407908 866115 292083 856202 225015 583502 210766 031265 206134 808795 676672 / 71 > 4459 [i]
- extracting embedded OOA [i] would yield OOA(4459, 116, S4, 4, 425), but