Best Known (44, 44+∞, s)-Nets in Base 4
(44, 44+∞, 56)-Net over F4 — Constructive and digital
Digital (44, m, 56)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (44, 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
(44, 44+∞, 75)-Net over F4 — Digital
Digital (44, m, 75)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (44, 74)-sequence over F4, using
- t-expansion [i] based on digital (40, 74)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 40 and N(F) ≥ 75, using
- t-expansion [i] based on digital (40, 74)-sequence over F4, using
(44, 44+∞, 146)-Net in Base 4 — Upper bound on s
There is no (44, m, 147)-net in base 4 for arbitrarily large m, because
- m-reduction [i] would yield (44, 583, 147)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4583, 147, S4, 4, 539), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 23051 690513 784546 883284 910989 995732 444940 266964 568283 187683 232057 444811 559896 924838 315587 511543 944826 441964 317214 158161 027441 577036 444982 721866 517519 971850 498531 047478 788985 769517 465821 697916 128314 609085 305029 299536 309598 547495 732306 708185 736722 850208 689297 249191 847019 272117 614688 010173 752935 240451 822820 747486 222137 478258 607515 184716 457444 710308 380672 / 15 > 4583 [i]
- extracting embedded OOA [i] would yield OOA(4583, 147, S4, 4, 539), but