Best Known (178, 178+19, s)-Nets in Base 3
(178, 178+19, 932084)-Net over F3 — Constructive and digital
Digital (178, 197, 932084)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (7, 16, 18)-net over F3, using
- 2 times m-reduction [i] based on digital (7, 18, 18)-net over F3, 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 (7, 16, 18)-net over F3, using
(178, 178+19, 2796219)-Net over F3 — Digital
Digital (178, 197, 2796219)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3197, 2796219, F3, 3, 19) (dual of [(2796219, 3), 8388460, 20]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(316, 18, F3, 3, 9) (dual of [(18, 3), 38, 10]-NRT-code), using
- extracting embedded OOA [i] based on digital (7, 16, 18)-net over F3, using
- 2 times m-reduction [i] based on digital (7, 18, 18)-net over F3, using
- extracting embedded OOA [i] based on digital (7, 16, 18)-net over F3, using
- linear OOA(3181, 2796201, F3, 3, 19) (dual of [(2796201, 3), 8388422, 20]-NRT-code), using
- OOA 3-folding [i] 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]
- OOA 3-folding [i] based on linear OA(3181, large, F3, 19) (dual of [large, large−181, 20]-code), using
- linear OOA(316, 18, F3, 3, 9) (dual of [(18, 3), 38, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(178, 178+19, large)-Net in Base 3 — Upper bound on s
There is no (178, 197, large)-net in base 3, because
- 17 times m-reduction [i] would yield (178, 180, large)-net in base 3, but