Best Known (211−19, 211, s)-Nets in Base 3
(211−19, 211, 932180)-Net over F3 — Constructive and digital
Digital (192, 211, 932180)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (21, 30, 114)-net over F3, using
- trace code for nets [i] based on digital (1, 10, 38)-net over F27, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 1 and N(F) ≥ 38, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- trace code for nets [i] based on digital (1, 10, 38)-net over F27, 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 (21, 30, 114)-net over F3, using
(211−19, 211, 4194456)-Net over F3 — Digital
Digital (192, 211, 4194456)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3211, 4194456, F3, 2, 19) (dual of [(4194456, 2), 8388701, 20]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(330, 155, F3, 2, 9) (dual of [(155, 2), 280, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(330, 155, F3, 9) (dual of [155, 125, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(330, 242, F3, 9) (dual of [242, 212, 10]-code), using
- the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(330, 242, F3, 9) (dual of [242, 212, 10]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(330, 155, F3, 9) (dual of [155, 125, 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(330, 155, F3, 2, 9) (dual of [(155, 2), 280, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(211−19, 211, large)-Net in Base 3 — Upper bound on s
There is no (192, 211, large)-net in base 3, because
- 17 times m-reduction [i] would yield (192, 194, large)-net in base 3, but