Best Known (237−44, 237, s)-Nets in Base 3
(237−44, 237, 896)-Net over F3 — Constructive and digital
Digital (193, 237, 896)-net over F3, using
- 3 times m-reduction [i] based on digital (193, 240, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 60, 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, 60, 224)-net over F81, using
(237−44, 237, 3920)-Net over F3 — Digital
Digital (193, 237, 3920)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3237, 3920, F3, 44) (dual of [3920, 3683, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(3237, 6581, F3, 44) (dual of [6581, 6344, 45]-code), using
- construction X applied to Ce(43) ⊂ Ce(40) [i] based on
- linear OA(3233, 6561, F3, 44) (dual of [6561, 6328, 45]-code), using an extension Ce(43) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [i]
- 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(34, 20, F3, 2) (dual of [20, 16, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(43) ⊂ Ce(40) [i] based on
- discarding factors / shortening the dual code based on linear OA(3237, 6581, F3, 44) (dual of [6581, 6344, 45]-code), using
(237−44, 237, 624748)-Net in Base 3 — Upper bound on s
There is no (193, 237, 624749)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 119603 706937 420405 090054 142796 421544 255441 133145 459844 301284 671507 493229 746026 790845 394581 237388 985125 101467 520657 > 3237 [i]