Best Known (174, 174+16, s)-Nets in Base 3
(174, 174+16, 1049125)-Net over F3 — Constructive and digital
Digital (174, 190, 1049125)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (31, 39, 550)-net over F3, using
- net defined by OOA [i] based on linear OOA(339, 550, F3, 8, 8) (dual of [(550, 8), 4361, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(339, 2200, F3, 8) (dual of [2200, 2161, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [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(322, 2187, F3, 5) (dual of [2187, 2165, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- OA 4-folding and stacking [i] based on linear OA(339, 2200, F3, 8) (dual of [2200, 2161, 9]-code), using
- net defined by OOA [i] based on linear OOA(339, 550, F3, 8, 8) (dual of [(550, 8), 4361, 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 (31, 39, 550)-net over F3, using
(174, 174+16, 8347646)-Net over F3 — Digital
Digital (174, 190, 8347646)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3190, 8347646, F3, 16) (dual of [8347646, 8347456, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3190, large, F3, 16) (dual of [large, large−190, 17]-code), using
- 39 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]
- 39 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(3190, large, F3, 16) (dual of [large, large−190, 17]-code), using
(174, 174+16, large)-Net in Base 3 — Upper bound on s
There is no (174, 190, large)-net in base 3, because
- 14 times m-reduction [i] would yield (174, 176, large)-net in base 3, but