Best Known (1, ∞, s)-Nets in Base 8
(1, ∞, 14)-Net over F8 — Constructive and digital
Digital (1, m, 14)-net over F8 for arbitrarily large m, using
- net from sequence [i] based on digital (1, 13)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 1 and N(F) ≥ 14, using
(1, ∞, 14)-Net over F8 — Upper bound on s (digital)
There is no digital (1, m, 15)-net over F8 for arbitrarily large m, because
- m-reduction [i] would yield digital (1, 9, 15)-net over F8, but
- extracting embedded orthogonal array [i] would yield linear OA(89, 15, F8, 8) (dual of [15, 6, 9]-code), but
- “MPa†bound on codes from Brouwer’s database [i]
- extracting embedded orthogonal array [i] would yield linear OA(89, 15, F8, 8) (dual of [15, 6, 9]-code), but
(1, ∞, 17)-Net in Base 8 — Upper bound on s
There is no (1, m, 18)-net in base 8 for arbitrarily large m, because
- m-reduction [i] would yield (1, 17, 18)-net in base 8, but
- extracting embedded OOA [i] would yield OOA(817, 18, S8, 2, 16), but
- the linear programming bound for OOAs shows that M ≥ 666532 744850 833408 / 289 > 817 [i]
- extracting embedded OOA [i] would yield OOA(817, 18, S8, 2, 16), but