Best Known (237−46, 237, s)-Nets in Base 3
(237−46, 237, 896)-Net over F3 — Constructive and digital
Digital (191, 237, 896)-net over F3, using
- 31 times duplication [i] based on digital (190, 236, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 59, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 59, 224)-net over F81, using
(237−46, 237, 3085)-Net over F3 — Digital
Digital (191, 237, 3085)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3237, 3085, F3, 46) (dual of [3085, 2848, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(3237, 3281, F3, 46) (dual of [3281, 3044, 47]-code), using
- an extension Ce(45) of the narrow-sense BCH-code C(I) with length 3280 | 38−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- discarding factors / shortening the dual code based on linear OA(3237, 3281, F3, 46) (dual of [3281, 3044, 47]-code), using
(237−46, 237, 388887)-Net in Base 3 — Upper bound on s
There is no (191, 237, 388888)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 119601 833032 729459 445536 992959 614631 220361 823430 835681 416260 322140 076779 335932 693642 979154 898017 399404 009234 784289 > 3237 [i]