Best Known (178, 218, s)-Nets in Base 3
(178, 218, 896)-Net over F3 — Constructive and digital
Digital (178, 218, 896)-net over F3, using
- 2 times m-reduction [i] based on digital (178, 220, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 55, 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, 55, 224)-net over F81, using
(178, 218, 3948)-Net over F3 — Digital
Digital (178, 218, 3948)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3218, 3948, F3, 40) (dual of [3948, 3730, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3218, 6595, F3, 40) (dual of [6595, 6377, 41]-code), using
- construction XX applied to Ce(39) ⊂ Ce(34) ⊂ Ce(33) [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(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(38, 33, F3, 4) (dual of [33, 25, 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
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(39) ⊂ Ce(34) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(3218, 6595, F3, 40) (dual of [6595, 6377, 41]-code), using
(178, 218, 658999)-Net in Base 3 — Upper bound on s
There is no (178, 218, 659000)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 102 905976 769037 971398 292075 129176 018130 288240 328438 819895 183195 182379 305443 707894 948648 247862 133271 102401 > 3218 [i]