Best Known (204, 204+19, s)-Nets in Base 3
(204, 204+19, 932612)-Net over F3 — Constructive and digital
Digital (204, 223, 932612)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (33, 42, 546)-net over F3, using
- net defined by OOA [i] based on linear OOA(342, 546, F3, 9, 9) (dual of [(546, 9), 4872, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(342, 2185, F3, 9) (dual of [2185, 2143, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(342, 2186, F3, 9) (dual of [2186, 2144, 10]-code), using
- 1 times truncation [i] based on linear OA(343, 2187, F3, 10) (dual of [2187, 2144, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- 1 times truncation [i] based on linear OA(343, 2187, F3, 10) (dual of [2187, 2144, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(342, 2186, F3, 9) (dual of [2186, 2144, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(342, 2185, F3, 9) (dual of [2185, 2143, 10]-code), using
- net defined by OOA [i] based on linear OOA(342, 546, F3, 9, 9) (dual of [(546, 9), 4872, 10]-NRT-code), 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 (33, 42, 546)-net over F3, using
(204, 204+19, 6103182)-Net over F3 — Digital
Digital (204, 223, 6103182)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3223, 6103182, F3, 19) (dual of [6103182, 6102959, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3223, large, F3, 19) (dual of [large, large−223, 20]-code), using
- 42 times code embedding in larger space [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]
- 42 times code embedding in larger space [i] based on linear OA(3181, large, F3, 19) (dual of [large, large−181, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3223, large, F3, 19) (dual of [large, large−223, 20]-code), using
(204, 204+19, large)-Net in Base 3 — Upper bound on s
There is no (204, 223, large)-net in base 3, because
- 17 times m-reduction [i] would yield (204, 206, large)-net in base 3, but