Best Known (217−19, 217, s)-Nets in Base 3
(217−19, 217, 932394)-Net over F3 — Constructive and digital
Digital (198, 217, 932394)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (27, 36, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 9, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- trace code for nets [i] based on digital (0, 9, 82)-net over F81, using
- digital (162, 181, 932066)-net over F3, using
- net defined by OOA [i] based on linear OOA(3181, 932066, F3, 19, 19) (dual of [(932066, 19), 17709073, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3181, 8388595, F3, 19) (dual of [8388595, 8388414, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3181, large, F3, 19) (dual of [large, large−181, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3181, large, F3, 19) (dual of [large, large−181, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3181, 8388595, F3, 19) (dual of [8388595, 8388414, 20]-code), using
- net defined by OOA [i] based on linear OOA(3181, 932066, F3, 19, 19) (dual of [(932066, 19), 17709073, 20]-NRT-code), using
- digital (27, 36, 328)-net over F3, using
(217−19, 217, 4194706)-Net over F3 — Digital
Digital (198, 217, 4194706)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3217, 4194706, F3, 2, 19) (dual of [(4194706, 2), 8389195, 20]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(336, 405, F3, 2, 9) (dual of [(405, 2), 774, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(336, 405, F3, 9) (dual of [405, 369, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(336, 728, F3, 9) (dual of [728, 692, 10]-code), using
- the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(336, 728, F3, 9) (dual of [728, 692, 10]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(336, 405, F3, 9) (dual of [405, 369, 10]-code), using
- linear OOA(3181, 4194301, F3, 2, 19) (dual of [(4194301, 2), 8388421, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3181, 8388602, F3, 19) (dual of [8388602, 8388421, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3181, large, F3, 19) (dual of [large, large−181, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3181, large, F3, 19) (dual of [large, large−181, 20]-code), using
- OOA 2-folding [i] based on linear OA(3181, 8388602, F3, 19) (dual of [8388602, 8388421, 20]-code), using
- linear OOA(336, 405, F3, 2, 9) (dual of [(405, 2), 774, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(217−19, 217, large)-Net in Base 3 — Upper bound on s
There is no (198, 217, large)-net in base 3, because
- 17 times m-reduction [i] would yield (198, 200, large)-net in base 3, but