Best Known (209, 250, s)-Nets in Base 3
(209, 250, 1488)-Net over F3 — Constructive and digital
Digital (209, 250, 1488)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 22, 8)-net over F3, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 2 and N(F) ≥ 8, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- digital (187, 228, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 57, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 57, 370)-net over F81, using
- digital (2, 22, 8)-net over F3, using
(209, 250, 9727)-Net over F3 — Digital
Digital (209, 250, 9727)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3250, 9727, F3, 2, 41) (dual of [(9727, 2), 19204, 42]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3250, 9858, F3, 2, 41) (dual of [(9858, 2), 19466, 42]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3250, 19716, F3, 41) (dual of [19716, 19466, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(36) [i] based on
- linear OA(3244, 19683, F3, 41) (dual of [19683, 19439, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(3217, 19683, F3, 37) (dual of [19683, 19466, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(36, 33, F3, 3) (dual of [33, 27, 4]-code or 33-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(40) ⊂ Ce(36) [i] based on
- OOA 2-folding [i] based on linear OA(3250, 19716, F3, 41) (dual of [19716, 19466, 42]-code), using
- discarding factors / shortening the dual code based on linear OOA(3250, 9858, F3, 2, 41) (dual of [(9858, 2), 19466, 42]-NRT-code), using
(209, 250, 3617708)-Net in Base 3 — Upper bound on s
There is no (209, 250, 3617709)-net in base 3, because
- 1 times m-reduction [i] would yield (209, 249, 3617709)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 63561 585299 954412 433075 766214 868333 997913 626373 451261 904662 584098 954365 936117 142146 275026 010156 482395 311753 392735 152977 > 3249 [i]