Best Known (1, 1+∞, s)-Nets in Base 4
(1, 1+∞, 9)-Net over F4 — Constructive and digital
Digital (1, m, 9)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- the Hermitian function field over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
(1, 1+∞, 9)-Net in Base 4 — Upper bound on s
There is no (1, m, 10)-net in base 4 for arbitrarily large m, because
- m-reduction [i] would yield (1, 9, 10)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(49, 10, S4, 2, 8), but
- the linear programming bound for OOAs shows that M ≥ 7 340032 / 27 > 49 [i]
- extracting embedded OOA [i] would yield OOA(49, 10, S4, 2, 8), but