Best Known (170−14, 170, s)-Nets in Base 3
(170−14, 170, 1366564)-Net over F3 — Constructive and digital
Digital (156, 170, 1366564)-net over F3, using
- trace code for nets [i] based on digital (71, 85, 683282)-net over F9, using
- net defined by OOA [i] based on linear OOA(985, 683282, F9, 14, 14) (dual of [(683282, 14), 9565863, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(985, 4782974, F9, 14) (dual of [4782974, 4782889, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(985, 4782976, F9, 14) (dual of [4782976, 4782891, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [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(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(90, 7, F9, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(985, 4782976, F9, 14) (dual of [4782976, 4782891, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(985, 4782974, F9, 14) (dual of [4782974, 4782889, 15]-code), using
- net defined by OOA [i] based on linear OOA(985, 683282, F9, 14, 14) (dual of [(683282, 14), 9565863, 15]-NRT-code), using
(170−14, 170, large)-Net over F3 — Digital
Digital (156, 170, large)-net over F3, using
- 35 times duplication [i] based on digital (151, 165, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3165, large, F3, 14) (dual of [large, large−165, 15]-code), using
- 29 times code embedding in larger space [i] based on linear OA(3136, large, F3, 14) (dual of [large, large−136, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 29 times code embedding in larger space [i] based on linear OA(3136, large, F3, 14) (dual of [large, large−136, 15]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3165, large, F3, 14) (dual of [large, large−165, 15]-code), using
(170−14, 170, large)-Net in Base 3 — Upper bound on s
There is no (156, 170, large)-net in base 3, because
- 12 times m-reduction [i] would yield (156, 158, large)-net in base 3, but