Best Known (189−16, 189, s)-Nets in Base 3
(189−16, 189, 1049124)-Net over F3 — Constructive and digital
Digital (173, 189, 1049124)-net over F3, using
- 31 times duplication [i] based on digital (172, 188, 1049124)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (29, 37, 549)-net over F3, using
- net defined by OOA [i] based on linear OOA(337, 549, F3, 8, 8) (dual of [(549, 8), 4355, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(337, 2196, F3, 8) (dual of [2196, 2159, 9]-code), using
- construction X4 applied to C([0,7]) ⊂ C([1,6]) [i] based on
- linear OA(336, 2186, F3, 8) (dual of [2186, 2150, 9]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(328, 2186, F3, 6) (dual of [2186, 2158, 7]-code), using the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(39, 10, F3, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,3)), using
- dual of repetition code with length 10 [i]
- linear OA(31, 10, F3, 1) (dual of [10, 9, 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 X4 applied to C([0,7]) ⊂ C([1,6]) [i] based on
- OA 4-folding and stacking [i] based on linear OA(337, 2196, F3, 8) (dual of [2196, 2159, 9]-code), using
- net defined by OOA [i] based on linear OOA(337, 549, F3, 8, 8) (dual of [(549, 8), 4355, 9]-NRT-code), using
- digital (135, 151, 1048575)-net over F3, using
- net defined by OOA [i] based on linear OOA(3151, 1048575, F3, 16, 16) (dual of [(1048575, 16), 16777049, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(3151, 8388600, F3, 16) (dual of [8388600, 8388449, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3151, large, F3, 16) (dual of [large, large−151, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(3151, large, F3, 16) (dual of [large, large−151, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(3151, 8388600, F3, 16) (dual of [8388600, 8388449, 17]-code), using
- net defined by OOA [i] based on linear OOA(3151, 1048575, F3, 16, 16) (dual of [(1048575, 16), 16777049, 17]-NRT-code), using
- digital (29, 37, 549)-net over F3, using
- (u, u+v)-construction [i] based on
(189−16, 189, 7717628)-Net over F3 — Digital
Digital (173, 189, 7717628)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3189, 7717628, F3, 16) (dual of [7717628, 7717439, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3189, large, F3, 16) (dual of [large, large−189, 17]-code), using
- 38 times code embedding in larger space [i] based on linear OA(3151, large, F3, 16) (dual of [large, large−151, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 38 times code embedding in larger space [i] based on linear OA(3151, large, F3, 16) (dual of [large, large−151, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3189, large, F3, 16) (dual of [large, large−189, 17]-code), using
(189−16, 189, large)-Net in Base 3 — Upper bound on s
There is no (173, 189, large)-net in base 3, because
- 14 times m-reduction [i] would yield (173, 175, large)-net in base 3, but