Best Known (186, 186+18, s)-Nets in Base 3
(186, 186+18, 932123)-Net over F3 — Constructive and digital
Digital (186, 204, 932123)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (15, 24, 56)-net over F3, using
- trace code for nets [i] based on digital (3, 12, 28)-net over F9, using
- net from sequence [i] based on digital (3, 27)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 3 and N(F) ≥ 28, using
- the Hermitian function field over F9 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 3 and N(F) ≥ 28, using
- net from sequence [i] based on digital (3, 27)-sequence over F9, using
- trace code for nets [i] based on digital (3, 12, 28)-net over F9, using
- digital (162, 180, 932067)-net over F3, using
- net defined by OOA [i] based on linear OOA(3180, 932067, F3, 18, 18) (dual of [(932067, 18), 16777026, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3180, large, F3, 18) (dual of [large, large−180, 19]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OA 9-folding and stacking [i] based on linear OA(3180, large, F3, 18) (dual of [large, large−180, 19]-code), using
- net defined by OOA [i] based on linear OOA(3180, 932067, F3, 18, 18) (dual of [(932067, 18), 16777026, 19]-NRT-code), using
- digital (15, 24, 56)-net over F3, using
(186, 186+18, 4194359)-Net over F3 — Digital
Digital (186, 204, 4194359)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3204, 4194359, F3, 2, 18) (dual of [(4194359, 2), 8388514, 19]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(324, 58, F3, 2, 9) (dual of [(58, 2), 92, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(324, 58, F3, 9) (dual of [58, 34, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(324, 80, F3, 9) (dual of [80, 56, 10]-code), using
- the primitive narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(324, 80, F3, 9) (dual of [80, 56, 10]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(324, 58, F3, 9) (dual of [58, 34, 10]-code), using
- linear OOA(3180, 4194301, F3, 2, 18) (dual of [(4194301, 2), 8388422, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3180, 8388602, F3, 18) (dual of [8388602, 8388422, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3180, large, F3, 18) (dual of [large, large−180, 19]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(3180, large, F3, 18) (dual of [large, large−180, 19]-code), using
- OOA 2-folding [i] based on linear OA(3180, 8388602, F3, 18) (dual of [8388602, 8388422, 19]-code), using
- linear OOA(324, 58, F3, 2, 9) (dual of [(58, 2), 92, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(186, 186+18, large)-Net in Base 3 — Upper bound on s
There is no (186, 204, large)-net in base 3, because
- 16 times m-reduction [i] would yield (186, 188, large)-net in base 3, but