Best Known (186, 226, s)-Nets in Base 3
(186, 226, 1480)-Net over F3 — Constructive and digital
Digital (186, 226, 1480)-net over F3, using
- 32 times duplication [i] based on digital (184, 224, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 56, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 56, 370)-net over F81, using
(186, 226, 4985)-Net over F3 — Digital
Digital (186, 226, 4985)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3226, 4985, F3, 40) (dual of [4985, 4759, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3226, 6618, F3, 40) (dual of [6618, 6392, 41]-code), using
- construction X applied to Ce(39) ⊂ Ce(31) [i] based on
- linear OA(3209, 6561, F3, 40) (dual of [6561, 6352, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(3169, 6561, F3, 32) (dual of [6561, 6392, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(317, 57, F3, 7) (dual of [57, 40, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(317, 80, F3, 7) (dual of [80, 63, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(317, 80, F3, 7) (dual of [80, 63, 8]-code), using
- construction X applied to Ce(39) ⊂ Ce(31) [i] based on
- discarding factors / shortening the dual code based on linear OA(3226, 6618, F3, 40) (dual of [6618, 6392, 41]-code), using
(186, 226, 1022676)-Net in Base 3 — Upper bound on s
There is no (186, 226, 1022677)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 675166 193070 156773 747794 550442 873557 883006 214683 909077 922152 845463 839432 696626 275726 689855 616352 282377 156241 > 3226 [i]