Best Known (191, 191+35, s)-Nets in Base 3
(191, 191+35, 1493)-Net over F3 — Constructive and digital
Digital (191, 226, 1493)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (5, 22, 13)-net over F3, using
- net from sequence [i] based on digital (5, 12)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F3 with g(F) = 4, N(F) = 12, and 1 place with degree 2 [i] based on function field F/F3 with g(F) = 4 and N(F) ≥ 12, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (5, 12)-sequence over F3, using
- digital (169, 204, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 51, 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, 51, 370)-net over F81, using
- digital (5, 22, 13)-net over F3, using
(191, 191+35, 11756)-Net over F3 — Digital
Digital (191, 226, 11756)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3226, 11756, F3, 35) (dual of [11756, 11530, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3226, 19746, F3, 35) (dual of [19746, 19520, 36]-code), using
- 3 times code embedding in larger space [i] based on linear OA(3223, 19743, F3, 35) (dual of [19743, 19520, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(27) [i] based on
- linear OA(3208, 19683, F3, 35) (dual of [19683, 19475, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(3163, 19683, F3, 28) (dual of [19683, 19520, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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(34) ⊂ Ce(27) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(3223, 19743, F3, 35) (dual of [19743, 19520, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3226, 19746, F3, 35) (dual of [19746, 19520, 36]-code), using
(191, 191+35, 7408911)-Net in Base 3 — Upper bound on s
There is no (191, 226, 7408912)-net in base 3, because
- 1 times m-reduction [i] would yield (191, 225, 7408912)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 225052 205274 251970 958768 418429 187498 919652 278280 731616 206459 205592 175303 932332 602938 831342 715520 320105 678881 > 3225 [i]