Best Known (189, 189+16, s)-Nets in Base 3
(189, 189+16, 1195748)-Net over F3 — Constructive and digital
Digital (189, 205, 1195748)-net over F3, using
- 31 times duplication [i] based on digital (188, 204, 1195748)-net over F3, using
- trace code for nets [i] based on digital (86, 102, 597874)-net over F9, using
- net defined by OOA [i] based on linear OOA(9102, 597874, F9, 16, 16) (dual of [(597874, 16), 9565882, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(9102, 4782992, F9, 16) (dual of [4782992, 4782890, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(9102, 4782993, F9, 16) (dual of [4782993, 4782891, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- linear OA(999, 4782969, F9, 16) (dual of [4782969, 4782870, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(978, 4782969, F9, 13) (dual of [4782969, 4782891, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(93, 24, F9, 2) (dual of [24, 21, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 92−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(9102, 4782993, F9, 16) (dual of [4782993, 4782891, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(9102, 4782992, F9, 16) (dual of [4782992, 4782890, 17]-code), using
- net defined by OOA [i] based on linear OOA(9102, 597874, F9, 16, 16) (dual of [(597874, 16), 9565882, 17]-NRT-code), using
- trace code for nets [i] based on digital (86, 102, 597874)-net over F9, using
(189, 189+16, large)-Net over F3 — Digital
Digital (189, 205, large)-net over F3, using
- 32 times duplication [i] based on digital (187, 203, large)-net over F3, using
- t-expansion [i] based on digital (186, 203, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3203, large, F3, 17) (dual of [large, large−203, 18]-code), using
- 37 times code embedding in larger space [i] based on linear OA(3166, large, F3, 17) (dual of [large, large−166, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 37 times code embedding in larger space [i] based on linear OA(3166, large, F3, 17) (dual of [large, large−166, 18]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3203, large, F3, 17) (dual of [large, large−203, 18]-code), using
- t-expansion [i] based on digital (186, 203, large)-net over F3, using
(189, 189+16, large)-Net in Base 3 — Upper bound on s
There is no (189, 205, large)-net in base 3, because
- 14 times m-reduction [i] would yield (189, 191, large)-net in base 3, but