Best Known (237−38, 237, s)-Nets in Base 3
(237−38, 237, 1488)-Net over F3 — Constructive and digital
Digital (199, 237, 1488)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 21, 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 (178, 216, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 54, 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, 54, 370)-net over F81, using
- digital (2, 21, 8)-net over F3, using
(237−38, 237, 9865)-Net over F3 — Digital
Digital (199, 237, 9865)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3237, 9865, F3, 2, 38) (dual of [(9865, 2), 19493, 39]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3237, 19730, F3, 38) (dual of [19730, 19493, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(31) [i] based on
- linear OA(3226, 19683, F3, 38) (dual of [19683, 19457, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [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(311, 47, F3, 5) (dual of [47, 36, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to Ce(37) ⊂ Ce(31) [i] based on
- OOA 2-folding [i] based on linear OA(3237, 19730, F3, 38) (dual of [19730, 19493, 39]-code), using
(237−38, 237, 3545219)-Net in Base 3 — Upper bound on s
There is no (199, 237, 3545220)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 119602 147635 196920 236728 873655 594157 345688 406522 703807 625584 114263 322133 485897 762731 158088 141596 199914 696650 717105 > 3237 [i]