Best Known (154, 154+34, s)-Nets in Base 3
(154, 154+34, 896)-Net over F3 — Constructive and digital
Digital (154, 188, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 47, 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
(154, 154+34, 3896)-Net over F3 — Digital
Digital (154, 188, 3896)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3188, 3896, F3, 34) (dual of [3896, 3708, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3188, 6602, F3, 34) (dual of [6602, 6414, 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, 41, F3, 5) (dual of [41, 30, 6]-code), using
- (u, u+v)-construction [i] based on
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- (u, u+v)-construction [i] based on
- construction X applied to Ce(33) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3188, 6602, F3, 34) (dual of [6602, 6414, 35]-code), using
(154, 154+34, 678116)-Net in Base 3 — Upper bound on s
There is no (154, 188, 678117)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 499804 078181 524414 861047 996348 719355 964810 687789 314042 482251 511308 836771 970914 316633 569675 > 3188 [i]