Best Known (157, 191, s)-Nets in Base 3
(157, 191, 896)-Net over F3 — Constructive and digital
Digital (157, 191, 896)-net over F3, using
- 1 times m-reduction [i] based on digital (157, 192, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 48, 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, 48, 224)-net over F81, using
(157, 191, 4322)-Net over F3 — Digital
Digital (157, 191, 4322)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3191, 4322, F3, 34) (dual of [4322, 4131, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3191, 6607, F3, 34) (dual of [6607, 6416, 35]-code), using
- 3 times code embedding in larger space [i] based on linear OA(3188, 6604, F3, 34) (dual of [6604, 6416, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(27) [i] based on
- linear OA(3177, 6561, F3, 34) (dual of [6561, 6384, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3145, 6561, F3, 28) (dual of [6561, 6416, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(311, 43, F3, 5) (dual of [43, 32, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to Ce(33) ⊂ Ce(27) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(3188, 6604, F3, 34) (dual of [6604, 6416, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3191, 6607, F3, 34) (dual of [6607, 6416, 35]-code), using
(157, 191, 823197)-Net in Base 3 — Upper bound on s
There is no (157, 191, 823198)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 13 494680 424503 791998 605706 909533 520992 647916 336838 346645 575156 527576 799234 242643 822669 637917 > 3191 [i]