Best Known (219−41, 219, s)-Nets in Base 3
(219−41, 219, 896)-Net over F3 — Constructive and digital
Digital (178, 219, 896)-net over F3, using
- 1 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
(219−41, 219, 3539)-Net over F3 — Digital
Digital (178, 219, 3539)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3219, 3539, F3, 41) (dual of [3539, 3320, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3219, 6572, F3, 41) (dual of [6572, 6353, 42]-code), using
- construction XX applied to Ce(40) ⊂ Ce(39) ⊂ Ce(37) [i] 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]
- 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(3201, 6561, F3, 38) (dual of [6561, 6360, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(30, 9, F3, 0) (dual of [9, 9, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(31, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(40) ⊂ Ce(39) ⊂ Ce(37) [i] based on
- discarding factors / shortening the dual code based on linear OA(3219, 6572, F3, 41) (dual of [6572, 6353, 42]-code), using
(219−41, 219, 658999)-Net in Base 3 — Upper bound on s
There is no (178, 219, 659000)-net in base 3, because
- 1 times m-reduction [i] would yield (178, 218, 659000)-net in base 3, but
- 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]