Best Known (1, ∞, s)-Nets in Base 7
(1, ∞, 9)-Net over F7 — Constructive and digital
Digital (1, m, 9)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (1, 8)-sequence over F7, using
(1, ∞, 13)-Net over F7 — Digital
Digital (1, m, 13)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (1, 12)-sequence over F7, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 1 and N(F) ≥ 13, using
(1, ∞, 13)-Net over F7 — Upper bound on s (digital)
There is no digital (1, m, 14)-net over F7 for arbitrarily large m, because
- m-reduction [i] would yield digital (1, 8, 14)-net over F7, but
- extracting embedded orthogonal array [i] would yield linear OA(78, 14, F7, 7) (dual of [14, 6, 8]-code), but
- “MPa†bound on codes from Brouwer’s database [i]
- extracting embedded orthogonal array [i] would yield linear OA(78, 14, F7, 7) (dual of [14, 6, 8]-code), but
(1, ∞, 15)-Net in Base 7 — Upper bound on s
There is no (1, m, 16)-net in base 7 for arbitrarily large m, because
- m-reduction [i] would yield (1, 15, 16)-net in base 7, but
- extracting embedded OOA [i] would yield OOA(715, 16, S7, 2, 14), but
- the linear programming bound for OOAs shows that M ≥ 365 562236 265611 / 75 > 715 [i]
- extracting embedded OOA [i] would yield OOA(715, 16, S7, 2, 14), but