Best Known (51, 51+19, s)-Nets in Base 3
(51, 51+19, 192)-Net over F3 — Constructive and digital
Digital (51, 70, 192)-net over F3, using
- 31 times duplication [i] based on digital (50, 69, 192)-net over F3, using
- trace code for nets [i] based on digital (4, 23, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 4 and N(F) ≥ 64, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- trace code for nets [i] based on digital (4, 23, 64)-net over F27, using
(51, 51+19, 296)-Net over F3 — Digital
Digital (51, 70, 296)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(370, 296, F3, 19) (dual of [296, 226, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(370, 364, F3, 19) (dual of [364, 294, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 364 | 36−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(370, 364, F3, 19) (dual of [364, 294, 20]-code), using
(51, 51+19, 9424)-Net in Base 3 — Upper bound on s
There is no (51, 70, 9425)-net in base 3, because
- 1 times m-reduction [i] would yield (51, 69, 9425)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 834 759327 379613 201172 269564 112771 > 369 [i]