Best Known (1, 1+7, s)-Nets in Base 5
(1, 1+7, 10)-Net over F5 — Constructive and digital
Digital (1, 8, 10)-net over F5, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 1 and N(F) ≥ 10, using
(1, 1+7, 11)-Net over F5 — Upper bound on s (digital)
There is no digital (1, 8, 12)-net over F5, because
- 1 times m-reduction [i] would yield digital (1, 7, 12)-net over F5, but
- extracting embedded orthogonal array [i] would yield linear OA(57, 12, F5, 6) (dual of [12, 5, 7]-code), but
(1, 1+7, 17)-Net in Base 5 — Upper bound on s
There is no (1, 8, 18)-net in base 5, because
- 1 times m-reduction [i] would yield (1, 7, 18)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 81385 > 57 [i]