Best Known (69−18, 69, s)-Nets in Base 3
(69−18, 69, 204)-Net over F3 — Constructive and digital
Digital (51, 69, 204)-net over F3, using
- trace code for nets [i] based on digital (5, 23, 68)-net over F27, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 5 and N(F) ≥ 68, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
(69−18, 69, 349)-Net over F3 — Digital
Digital (51, 69, 349)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(369, 349, F3, 18) (dual of [349, 280, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(369, 364, F3, 18) (dual of [364, 295, 19]-code), using
- the narrow-sense BCH-code C(I) with length 364 | 36−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(369, 364, F3, 18) (dual of [364, 295, 19]-code), using
(69−18, 69, 9424)-Net in Base 3 — Upper bound on s
There is no (51, 69, 9425)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 834 759327 379613 201172 269564 112771 > 369 [i]