Best Known (171, 171+39, s)-Nets in Base 3
(171, 171+39, 896)-Net over F3 — Constructive and digital
Digital (171, 210, 896)-net over F3, using
- 32 times duplication [i] based on digital (169, 208, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 52, 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, 52, 224)-net over F81, using
(171, 171+39, 3595)-Net over F3 — Digital
Digital (171, 210, 3595)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3210, 3595, F3, 39) (dual of [3595, 3385, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3210, 6579, F3, 39) (dual of [6579, 6369, 40]-code), using
- construction X applied to C([0,19]) ⊂ C([0,18]) [i] based on
- linear OA(3209, 6562, F3, 39) (dual of [6562, 6353, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- linear OA(3193, 6562, F3, 37) (dual of [6562, 6369, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(31, 17, F3, 1) (dual of [17, 16, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,19]) ⊂ C([0,18]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3210, 6579, F3, 39) (dual of [6579, 6369, 40]-code), using
(171, 171+39, 702276)-Net in Base 3 — Upper bound on s
There is no (171, 210, 702277)-net in base 3, because
- 1 times m-reduction [i] would yield (171, 209, 702277)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 5228 211180 483822 644205 992035 939119 304670 434471 382172 838799 849822 069378 222041 752344 191819 518041 663019 > 3209 [i]