Best Known (211, 250, s)-Nets in Base 3
(211, 250, 1500)-Net over F3 — Constructive and digital
Digital (211, 250, 1500)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (11, 30, 20)-net over F3, using
- net from sequence [i] based on digital (11, 19)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F3 with g(F) = 9, N(F) = 19, and 1 place with degree 3 [i] based on function field F/F3 with g(F) = 9 and N(F) ≥ 19, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (11, 19)-sequence over F3, using
- digital (181, 220, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 55, 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, 55, 370)-net over F81, using
- digital (11, 30, 20)-net over F3, using
(211, 250, 11871)-Net over F3 — Digital
Digital (211, 250, 11871)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3250, 11871, F3, 39) (dual of [11871, 11621, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3250, 19743, F3, 39) (dual of [19743, 19493, 40]-code), using
- construction X applied to Ce(39) ⊂ Ce(31) [i] based on
- linear OA(3235, 19683, F3, 40) (dual of [19683, 19448, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(3190, 19683, F3, 32) (dual of [19683, 19493, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(315, 60, F3, 6) (dual of [60, 45, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(315, 85, F3, 6) (dual of [85, 70, 7]-code), using
- construction X applied to Ce(39) ⊂ Ce(31) [i] based on
- discarding factors / shortening the dual code based on linear OA(3250, 19743, F3, 39) (dual of [19743, 19493, 40]-code), using
(211, 250, 7095515)-Net in Base 3 — Upper bound on s
There is no (211, 250, 7095516)-net in base 3, because
- 1 times m-reduction [i] would yield (211, 249, 7095516)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 63561 260443 066517 739802 404963 256517 360629 093310 598178 593031 370959 527188 941580 914251 250619 766315 572815 380656 051278 504465 > 3249 [i]