Best Known (216−40, 216, s)-Nets in Base 3
(216−40, 216, 896)-Net over F3 — Constructive and digital
Digital (176, 216, 896)-net over F3, using
- t-expansion [i] based on digital (175, 216, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 54, 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, 54, 224)-net over F81, using
(216−40, 216, 3725)-Net over F3 — Digital
Digital (176, 216, 3725)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3216, 3725, F3, 40) (dual of [3725, 3509, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3216, 6588, F3, 40) (dual of [6588, 6372, 41]-code), using
- construction X applied to Ce(39) ⊂ Ce(34) [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(3185, 6561, F3, 35) (dual of [6561, 6376, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(37, 27, F3, 4) (dual of [27, 20, 5]-code), using
- an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 26 = 33−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- construction X applied to Ce(39) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(3216, 6588, F3, 40) (dual of [6588, 6372, 41]-code), using
(216−40, 216, 590434)-Net in Base 3 — Upper bound on s
There is no (176, 216, 590435)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 11 434153 779256 846829 498165 331082 719092 666324 960399 017857 615301 290811 374109 371400 688799 513140 870114 693817 > 3216 [i]