Best Known (19−15, 19, s)-Nets in Base 5
(19−15, 19, 18)-Net over F5 — Constructive and digital
Digital (4, 19, 18)-net over F5, using
- net from sequence [i] based on digital (4, 17)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 4 and N(F) ≥ 18, using
(19−15, 19, 41)-Net over F5 — Upper bound on s (digital)
There is no digital (4, 19, 42)-net over F5, because
- extracting embedded orthogonal array [i] would yield linear OA(519, 42, F5, 15) (dual of [42, 23, 16]-code), but
- construction Y1 [i] would yield
- linear OA(518, 23, F5, 15) (dual of [23, 5, 16]-code), but
- residual code [i] would yield OA(53, 7, S5, 3), but
- OA(523, 42, S5, 19), but
- discarding factors would yield OA(523, 41, S5, 19), but
- the linear programming bound shows that M ≥ 1431 381081 044673 919677 734375 / 117616 346784 > 523 [i]
- discarding factors would yield OA(523, 41, S5, 19), but
- linear OA(518, 23, F5, 15) (dual of [23, 5, 16]-code), but
- construction Y1 [i] would yield
(19−15, 19, 48)-Net in Base 5 — Upper bound on s
There is no (4, 19, 49)-net in base 5, because
- 1 times m-reduction [i] would yield (4, 18, 49)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 4 129466 784493 > 518 [i]