Best Known (224−19, 224, s)-Nets in Base 3
(224−19, 224, 932614)-Net over F3 — Constructive and digital
Digital (205, 224, 932614)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (34, 43, 548)-net over F3, using
- net defined by OOA [i] based on linear OOA(343, 548, F3, 9, 9) (dual of [(548, 9), 4889, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(343, 2193, F3, 9) (dual of [2193, 2150, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(343, 2194, F3, 9) (dual of [2194, 2151, 10]-code), using
- 1 times truncation [i] based on linear OA(344, 2195, F3, 10) (dual of [2195, 2151, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [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]
- linear OA(336, 2187, F3, 8) (dual of [2187, 2151, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(31, 8, F3, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- 1 times truncation [i] based on linear OA(344, 2195, F3, 10) (dual of [2195, 2151, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(343, 2194, F3, 9) (dual of [2194, 2151, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(343, 2193, F3, 9) (dual of [2193, 2150, 10]-code), using
- net defined by OOA [i] based on linear OOA(343, 548, F3, 9, 9) (dual of [(548, 9), 4889, 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 (34, 43, 548)-net over F3, using
(224−19, 224, 6510620)-Net over F3 — Digital
Digital (205, 224, 6510620)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3224, 6510620, F3, 19) (dual of [6510620, 6510396, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3224, large, F3, 19) (dual of [large, large−224, 20]-code), using
- 43 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]
- 43 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(3224, large, F3, 19) (dual of [large, large−224, 20]-code), using
(224−19, 224, large)-Net in Base 3 — Upper bound on s
There is no (205, 224, large)-net in base 3, because
- 17 times m-reduction [i] would yield (205, 207, large)-net in base 3, but