Best Known (218−17, 218, s)-Nets in Base 3
(218−17, 218, 1195748)-Net over F3 — Constructive and digital
Digital (201, 218, 1195748)-net over F3, using
- trace code for nets [i] based on digital (92, 109, 597874)-net over F9, using
- net defined by OOA [i] based on linear OOA(9109, 597874, F9, 17, 17) (dual of [(597874, 17), 10163749, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(9109, 4782993, F9, 17) (dual of [4782993, 4782884, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(9106, 4782969, F9, 17) (dual of [4782969, 4782863, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(93, 24, F9, 2) (dual of [24, 21, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 92−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- OOA 8-folding and stacking with additional row [i] based on linear OA(9109, 4782993, F9, 17) (dual of [4782993, 4782884, 18]-code), using
- net defined by OOA [i] based on linear OOA(9109, 597874, F9, 17, 17) (dual of [(597874, 17), 10163749, 18]-NRT-code), using
(218−17, 218, large)-Net over F3 — Digital
Digital (201, 218, large)-net over F3, using
- 32 times duplication [i] based on digital (199, 216, large)-net over F3, using
- t-expansion [i] based on digital (198, 216, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3216, large, F3, 18) (dual of [large, large−216, 19]-code), using
- 36 times code embedding in larger space [i] based on linear OA(3180, large, F3, 18) (dual of [large, large−180, 19]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 36 times code embedding in larger space [i] based on linear OA(3180, large, F3, 18) (dual of [large, large−180, 19]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3216, large, F3, 18) (dual of [large, large−216, 19]-code), using
- t-expansion [i] based on digital (198, 216, large)-net over F3, using
(218−17, 218, large)-Net in Base 3 — Upper bound on s
There is no (201, 218, large)-net in base 3, because
- 15 times m-reduction [i] would yield (201, 203, large)-net in base 3, but