Best Known (214−40, 214, s)-Nets in Base 3
(214−40, 214, 896)-Net over F3 — Constructive and digital
Digital (174, 214, 896)-net over F3, using
- 32 times duplication [i] based on digital (172, 212, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 53, 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, 53, 224)-net over F81, using
(214−40, 214, 3513)-Net over F3 — Digital
Digital (174, 214, 3513)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3214, 3513, F3, 40) (dual of [3513, 3299, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3214, 6582, F3, 40) (dual of [6582, 6368, 41]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3213, 6581, F3, 40) (dual of [6581, 6368, 41]-code), using
- construction X applied to Ce(39) ⊂ Ce(36) [i] based on
- linear OA(3209, 6561, F3, 40) (dual of [6561, 6352, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- 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(34, 20, F3, 2) (dual of [20, 16, 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(39) ⊂ Ce(36) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3213, 6581, F3, 40) (dual of [6581, 6368, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3214, 6582, F3, 40) (dual of [6582, 6368, 41]-code), using
(214−40, 214, 529002)-Net in Base 3 — Upper bound on s
There is no (174, 214, 529003)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1 270451 709527 423819 731214 548735 462600 886111 387044 966818 202882 719529 498182 091466 587954 919789 585527 158201 > 3214 [i]