Best Known (221−41, 221, s)-Nets in Base 3
(221−41, 221, 896)-Net over F3 — Constructive and digital
Digital (180, 221, 896)-net over F3, using
- 31 times duplication [i] based on digital (179, 220, 896)-net over F3, using
- t-expansion [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
- t-expansion [i] based on digital (178, 220, 896)-net over F3, using
(221−41, 221, 3746)-Net over F3 — Digital
Digital (180, 221, 3746)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3221, 3746, F3, 41) (dual of [3746, 3525, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3221, 6581, F3, 41) (dual of [6581, 6360, 42]-code), using
- construction X applied to Ce(40) ⊂ 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(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(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(40) ⊂ Ce(37) [i] based on
- discarding factors / shortening the dual code based on linear OA(3221, 6581, F3, 41) (dual of [6581, 6360, 42]-code), using
(221−41, 221, 735526)-Net in Base 3 — Upper bound on s
There is no (180, 221, 735527)-net in base 3, because
- 1 times m-reduction [i] would yield (180, 220, 735527)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 926 142810 381595 018933 088665 235004 745724 463448 859238 366831 925289 243113 231973 040990 528452 721791 824493 039961 > 3220 [i]