Best Known (187−16, 187, s)-Nets in Base 3
(187−16, 187, 1049123)-Net over F3 — Constructive and digital
Digital (171, 187, 1049123)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (28, 36, 548)-net over F3, using
- net defined by OOA [i] based on linear OOA(336, 548, F3, 8, 8) (dual of [(548, 8), 4348, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(336, 2192, F3, 8) (dual of [2192, 2156, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(336, 2194, F3, 8) (dual of [2194, 2158, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- 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(329, 2187, F3, 7) (dual of [2187, 2158, 8]-code), using an extension Ce(6) of 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(30, 7, F3, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(336, 2194, F3, 8) (dual of [2194, 2158, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(336, 2192, F3, 8) (dual of [2192, 2156, 9]-code), using
- net defined by OOA [i] based on linear OOA(336, 548, F3, 8, 8) (dual of [(548, 8), 4348, 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 (28, 36, 548)-net over F3, using
(187−16, 187, 6596652)-Net over F3 — Digital
Digital (171, 187, 6596652)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3187, 6596652, F3, 16) (dual of [6596652, 6596465, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3187, large, F3, 16) (dual of [large, large−187, 17]-code), using
- 36 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]
- 36 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(3187, large, F3, 16) (dual of [large, large−187, 17]-code), using
(187−16, 187, large)-Net in Base 3 — Upper bound on s
There is no (171, 187, large)-net in base 3, because
- 14 times m-reduction [i] would yield (171, 173, large)-net in base 3, but