Best Known (199, 199+46, s)-Nets in Base 3
(199, 199+46, 896)-Net over F3 — Constructive and digital
Digital (199, 245, 896)-net over F3, using
- 3 times m-reduction [i] based on digital (199, 248, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 62, 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, 62, 224)-net over F81, using
(199, 199+46, 3775)-Net over F3 — Digital
Digital (199, 245, 3775)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3245, 3775, F3, 46) (dual of [3775, 3530, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(3245, 6581, F3, 46) (dual of [6581, 6336, 47]-code), using
- construction X applied to Ce(45) ⊂ Ce(42) [i] based on
- linear OA(3241, 6561, F3, 46) (dual of [6561, 6320, 47]-code), using an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(3225, 6561, F3, 43) (dual of [6561, 6336, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(34, 20, F3, 2) (dual of [20, 16, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(45) ⊂ Ce(42) [i] based on
- discarding factors / shortening the dual code based on linear OA(3245, 6581, F3, 46) (dual of [6581, 6336, 47]-code), using
(199, 199+46, 569885)-Net in Base 3 — Upper bound on s
There is no (199, 245, 569886)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 784 717894 249529 998813 738347 013783 744740 254709 994994 582101 730139 924351 541687 303320 762397 724733 496845 454468 905526 774553 > 3245 [i]