Best Known (218−39, 218, s)-Nets in Base 3
(218−39, 218, 896)-Net over F3 — Constructive and digital
Digital (179, 218, 896)-net over F3, using
- t-expansion [i] based on 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
- 2 times m-reduction [i] based on digital (178, 220, 896)-net over F3, using
(218−39, 218, 4569)-Net over F3 — Digital
Digital (179, 218, 4569)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3218, 4569, F3, 39) (dual of [4569, 4351, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3218, 6602, F3, 39) (dual of [6602, 6384, 40]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3217, 6601, F3, 39) (dual of [6601, 6384, 40]-code), using
- construction X applied to Ce(39) ⊂ 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(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, 40, F3, 4) (dual of [40, 32, 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
- construction X applied to Ce(39) ⊂ Ce(33) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3217, 6601, F3, 39) (dual of [6601, 6384, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3218, 6602, F3, 39) (dual of [6602, 6384, 40]-code), using
(218−39, 218, 1115334)-Net in Base 3 — Upper bound on s
There is no (179, 218, 1115335)-net in base 3, because
- 1 times m-reduction [i] would yield (179, 217, 1115335)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 34 301526 674397 850688 968504 467897 398175 328868 078096 272014 656493 420245 673584 800732 357180 707197 055200 139771 > 3217 [i]