Best Known (178, 178+15, s)-Nets in Base 3
(178, 178+15, 1366573)-Net over F3 — Constructive and digital
Digital (178, 193, 1366573)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 9, 9)-net over F3, using
- digital (169, 184, 1366564)-net over F3, using
- trace code for nets [i] based on digital (77, 92, 683282)-net over F9, using
- net defined by OOA [i] based on linear OOA(992, 683282, F9, 15, 15) (dual of [(683282, 15), 10249138, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(992, 4782975, F9, 15) (dual of [4782975, 4782883, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(992, 4782976, F9, 15) (dual of [4782976, 4782884, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [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(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(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(14) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(992, 4782976, F9, 15) (dual of [4782976, 4782884, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(992, 4782975, F9, 15) (dual of [4782975, 4782883, 16]-code), using
- net defined by OOA [i] based on linear OOA(992, 683282, F9, 15, 15) (dual of [(683282, 15), 10249138, 16]-NRT-code), using
- trace code for nets [i] based on digital (77, 92, 683282)-net over F9, using
(178, 178+15, large)-Net over F3 — Digital
Digital (178, 193, large)-net over F3, using
- 32 times duplication [i] based on digital (176, 191, large)-net over F3, using
- t-expansion [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
- t-expansion [i] based on digital (175, 191, large)-net over F3, using
(178, 178+15, large)-Net in Base 3 — Upper bound on s
There is no (178, 193, large)-net in base 3, because
- 13 times m-reduction [i] would yield (178, 180, large)-net in base 3, but