Best Known (191−40, 191, s)-Nets in Base 3
(191−40, 191, 688)-Net over F3 — Constructive and digital
Digital (151, 191, 688)-net over F3, using
- 1 times m-reduction [i] based on digital (151, 192, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 48, 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, 48, 172)-net over F81, using
(191−40, 191, 1790)-Net over F3 — Digital
Digital (151, 191, 1790)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3191, 1790, F3, 40) (dual of [1790, 1599, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3191, 2216, F3, 40) (dual of [2216, 2025, 41]-code), using
- construction X applied to Ce(39) ⊂ Ce(34) [i] based on
- linear OA(3183, 2187, F3, 40) (dual of [2187, 2004, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(3162, 2187, F3, 35) (dual of [2187, 2025, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(38, 29, F3, 4) (dual of [29, 21, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(39) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(3191, 2216, F3, 40) (dual of [2216, 2025, 41]-code), using
(191−40, 191, 149529)-Net in Base 3 — Upper bound on s
There is no (151, 191, 149530)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 13 495051 362139 205949 419205 869223 557560 312293 226486 578380 841458 421678 406296 920255 228379 485001 > 3191 [i]