Best Known (163, 177, s)-Nets in Base 3
(163, 177, 1366568)-Net over F3 — Constructive and digital
Digital (163, 177, 1366568)-net over F3, using
- 31 times duplication [i] based on digital (162, 176, 1366568)-net over F3, using
- trace code for nets [i] based on digital (74, 88, 683284)-net over F9, using
- net defined by OOA [i] based on linear OOA(988, 683284, F9, 14, 14) (dual of [(683284, 14), 9565888, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(988, 4782988, F9, 14) (dual of [4782988, 4782900, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(988, 4782993, F9, 14) (dual of [4782993, 4782905, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- linear OA(985, 4782969, F9, 14) (dual of [4782969, 4782884, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(964, 4782969, F9, 11) (dual of [4782969, 4782905, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(13) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(988, 4782993, F9, 14) (dual of [4782993, 4782905, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(988, 4782988, F9, 14) (dual of [4782988, 4782900, 15]-code), using
- net defined by OOA [i] based on linear OOA(988, 683284, F9, 14, 14) (dual of [(683284, 14), 9565888, 15]-NRT-code), using
- trace code for nets [i] based on digital (74, 88, 683284)-net over F9, using
(163, 177, large)-Net over F3 — Digital
Digital (163, 177, large)-net over F3, using
- 1 times m-reduction [i] based on digital (163, 178, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3178, large, F3, 15) (dual of [large, large−178, 16]-code), using
- 28 times code embedding in larger space [i] based on linear OA(3150, large, F3, 15) (dual of [large, large−150, 16]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- 28 times code embedding in larger space [i] based on linear OA(3150, large, F3, 15) (dual of [large, large−150, 16]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3178, large, F3, 15) (dual of [large, large−178, 16]-code), using
(163, 177, large)-Net in Base 3 — Upper bound on s
There is no (163, 177, large)-net in base 3, because
- 12 times m-reduction [i] would yield (163, 165, large)-net in base 3, but