Best Known (159, 159+14, s)-Nets in Base 3
(159, 159+14, 1366566)-Net over F3 — Constructive and digital
Digital (159, 173, 1366566)-net over F3, using
- 31 times duplication [i] based on digital (158, 172, 1366566)-net over F3, using
- trace code for nets [i] based on digital (72, 86, 683283)-net over F9, using
- net defined by OOA [i] based on linear OOA(986, 683283, F9, 14, 14) (dual of [(683283, 14), 9565876, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(986, 4782981, F9, 14) (dual of [4782981, 4782895, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(986, 4782984, F9, 14) (dual of [4782984, 4782898, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [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(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(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(13) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(986, 4782984, F9, 14) (dual of [4782984, 4782898, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(986, 4782981, F9, 14) (dual of [4782981, 4782895, 15]-code), using
- net defined by OOA [i] based on linear OOA(986, 683283, F9, 14, 14) (dual of [(683283, 14), 9565876, 15]-NRT-code), using
- trace code for nets [i] based on digital (72, 86, 683283)-net over F9, using
(159, 159+14, large)-Net over F3 — Digital
Digital (159, 173, large)-net over F3, using
- 38 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
(159, 159+14, large)-Net in Base 3 — Upper bound on s
There is no (159, 173, large)-net in base 3, because
- 12 times m-reduction [i] would yield (159, 161, large)-net in base 3, but