Best Known (191, 191+52, s)-Nets in Base 3
(191, 191+52, 688)-Net over F3 — Constructive and digital
Digital (191, 243, 688)-net over F3, using
- t-expansion [i] based on digital (190, 243, 688)-net over F3, using
- 1 times m-reduction [i] based on digital (190, 244, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 61, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 61, 172)-net over F81, using
- 1 times m-reduction [i] based on digital (190, 244, 688)-net over F3, using
(191, 191+52, 1940)-Net over F3 — Digital
Digital (191, 243, 1940)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3243, 1940, F3, 52) (dual of [1940, 1697, 53]-code), using
- discarding factors / shortening the dual code based on linear OA(3243, 2205, F3, 52) (dual of [2205, 1962, 53]-code), using
- construction X applied to Ce(51) ⊂ Ce(48) [i] based on
- linear OA(3239, 2187, F3, 52) (dual of [2187, 1948, 53]-code), using an extension Ce(51) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,51], and designed minimum distance d ≥ |I|+1 = 52 [i]
- linear OA(3225, 2187, F3, 49) (dual of [2187, 1962, 50]-code), using an extension Ce(48) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,48], and designed minimum distance d ≥ |I|+1 = 49 [i]
- linear OA(34, 18, F3, 2) (dual of [18, 14, 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(51) ⊂ Ce(48) [i] based on
- discarding factors / shortening the dual code based on linear OA(3243, 2205, F3, 52) (dual of [2205, 1962, 53]-code), using
(191, 191+52, 151858)-Net in Base 3 — Upper bound on s
There is no (191, 243, 151859)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 87 198814 220332 150722 728725 460600 915354 300756 489051 898432 906329 797803 892394 562779 662740 199289 909535 108472 688763 007509 > 3243 [i]