Best Known (58−16, 58, s)-Nets in Base 3
(58−16, 58, 156)-Net over F3 — Constructive and digital
Digital (42, 58, 156)-net over F3, using
- 31 times duplication [i] based on digital (41, 57, 156)-net over F3, using
- trace code for nets [i] based on digital (3, 19, 52)-net over F27, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 3 and N(F) ≥ 52, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
- trace code for nets [i] based on digital (3, 19, 52)-net over F27, using
(58−16, 58, 253)-Net over F3 — Digital
Digital (42, 58, 253)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(358, 253, F3, 16) (dual of [253, 195, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(358, 364, F3, 16) (dual of [364, 306, 17]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 364 | 36−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(358, 364, F3, 16) (dual of [364, 306, 17]-code), using
(58−16, 58, 5409)-Net in Base 3 — Upper bound on s
There is no (42, 58, 5410)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 4710 944707 318469 960892 186705 > 358 [i]