Best Known (175, 175+15, s)-Nets in Base 3
(175, 175+15, 1366568)-Net over F3 — Constructive and digital
Digital (175, 190, 1366568)-net over F3, using
- trace code for nets [i] based on digital (80, 95, 683284)-net over F9, using
- net defined by OOA [i] based on linear OOA(995, 683284, F9, 15, 15) (dual of [(683284, 15), 10249165, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(995, 4782989, F9, 15) (dual of [4782989, 4782894, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(995, 4782993, F9, 15) (dual of [4782993, 4782898, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- linear OA(992, 4782969, F9, 15) (dual of [4782969, 4782877, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(971, 4782969, F9, 12) (dual of [4782969, 4782898, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(14) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(995, 4782993, F9, 15) (dual of [4782993, 4782898, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(995, 4782989, F9, 15) (dual of [4782989, 4782894, 16]-code), using
- net defined by OOA [i] based on linear OOA(995, 683284, F9, 15, 15) (dual of [(683284, 15), 10249165, 16]-NRT-code), using
(175, 175+15, large)-Net over F3 — Digital
Digital (175, 190, large)-net over F3, using
- 1 times m-reduction [i] based on digital (175, 191, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3191, large, F3, 16) (dual of [large, large−191, 17]-code), using
- 40 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]
- 40 times code embedding in larger space [i] based on linear OA(3151, large, F3, 16) (dual of [large, large−151, 17]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3191, large, F3, 16) (dual of [large, large−191, 17]-code), using
(175, 175+15, large)-Net in Base 3 — Upper bound on s
There is no (175, 190, large)-net in base 3, because
- 13 times m-reduction [i] would yield (175, 177, large)-net in base 3, but