Best Known (176, 176+41, s)-Nets in Base 3
(176, 176+41, 896)-Net over F3 — Constructive and digital
Digital (176, 217, 896)-net over F3, using
- 31 times duplication [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
(176, 176+41, 3343)-Net over F3 — Digital
Digital (176, 217, 3343)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3217, 3343, F3, 41) (dual of [3343, 3126, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3217, 6561, F3, 41) (dual of [6561, 6344, 42]-code), using
- an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- discarding factors / shortening the dual code based on linear OA(3217, 6561, F3, 41) (dual of [6561, 6344, 42]-code), using
(176, 176+41, 590434)-Net in Base 3 — Upper bound on s
There is no (176, 217, 590435)-net in base 3, because
- 1 times m-reduction [i] would yield (176, 216, 590435)-net in base 3, but
- 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]