Best Known (174, 174+15, s)-Nets in Base 3
(174, 174+15, 1366566)-Net over F3 — Constructive and digital
Digital (174, 189, 1366566)-net over F3, using
- 33 times duplication [i] based on digital (171, 186, 1366566)-net over F3, using
- trace code for nets [i] based on digital (78, 93, 683283)-net over F9, using
- net defined by OOA [i] based on linear OOA(993, 683283, F9, 15, 15) (dual of [(683283, 15), 10249152, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(993, 4782982, F9, 15) (dual of [4782982, 4782889, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(993, 4782984, F9, 15) (dual of [4782984, 4782891, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [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(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(91, 15, F9, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(993, 4782984, F9, 15) (dual of [4782984, 4782891, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(993, 4782982, F9, 15) (dual of [4782982, 4782889, 16]-code), using
- net defined by OOA [i] based on linear OOA(993, 683283, F9, 15, 15) (dual of [(683283, 15), 10249152, 16]-NRT-code), using
- trace code for nets [i] based on digital (78, 93, 683283)-net over F9, using
(174, 174+15, large)-Net over F3 — Digital
Digital (174, 189, large)-net over F3, using
- 311 times duplication [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
(174, 174+15, large)-Net in Base 3 — Upper bound on s
There is no (174, 189, large)-net in base 3, because
- 13 times m-reduction [i] would yield (174, 176, large)-net in base 3, but