Best Known (199−36, 199, s)-Nets in Base 3
(199−36, 199, 896)-Net over F3 — Constructive and digital
Digital (163, 199, 896)-net over F3, using
- 1 times m-reduction [i] based on digital (163, 200, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 50, 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, 50, 224)-net over F81, using
(199−36, 199, 4032)-Net over F3 — Digital
Digital (163, 199, 4032)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3199, 4032, F3, 36) (dual of [4032, 3833, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3199, 6591, F3, 36) (dual of [6591, 6392, 37]-code), using
- construction X applied to Ce(36) ⊂ Ce(31) [i] based on
- linear OA(3193, 6561, F3, 37) (dual of [6561, 6368, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3169, 6561, F3, 32) (dual of [6561, 6392, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(36, 30, F3, 3) (dual of [30, 24, 4]-code or 30-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(36) ⊂ Ce(31) [i] based on
- discarding factors / shortening the dual code based on linear OA(3199, 6591, F3, 36) (dual of [6591, 6392, 37]-code), using
(199−36, 199, 711098)-Net in Base 3 — Upper bound on s
There is no (163, 199, 711099)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 88538 569678 982973 291598 044002 193560 341101 467324 678512 728803 934186 560975 763831 658517 850563 026165 > 3199 [i]