Best Known (218−29, 218, s)-Nets in Base 3
(218−29, 218, 12656)-Net over F3 — Constructive and digital
Digital (189, 218, 12656)-net over F3, using
- 32 times duplication [i] based on digital (187, 216, 12656)-net over F3, using
- net defined by OOA [i] based on linear OOA(3216, 12656, F3, 29, 29) (dual of [(12656, 29), 366808, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3216, 177185, F3, 29) (dual of [177185, 176969, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3216, 177186, F3, 29) (dual of [177186, 176970, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- linear OA(3210, 177147, F3, 29) (dual of [177147, 176937, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3177, 177147, F3, 25) (dual of [177147, 176970, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(36, 39, F3, 3) (dual of [39, 33, 4]-code or 39-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(3216, 177186, F3, 29) (dual of [177186, 176970, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3216, 177185, F3, 29) (dual of [177185, 176969, 30]-code), using
- net defined by OOA [i] based on linear OOA(3216, 12656, F3, 29, 29) (dual of [(12656, 29), 366808, 30]-NRT-code), using
(218−29, 218, 59062)-Net over F3 — Digital
Digital (189, 218, 59062)-net over F3, using
- 32 times duplication [i] based on digital (187, 216, 59062)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3216, 59062, F3, 3, 29) (dual of [(59062, 3), 176970, 30]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3216, 177186, F3, 29) (dual of [177186, 176970, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- linear OA(3210, 177147, F3, 29) (dual of [177147, 176937, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3177, 177147, F3, 25) (dual of [177147, 176970, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(36, 39, F3, 3) (dual of [39, 33, 4]-code or 39-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- OOA 3-folding [i] based on linear OA(3216, 177186, F3, 29) (dual of [177186, 176970, 30]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3216, 59062, F3, 3, 29) (dual of [(59062, 3), 176970, 30]-NRT-code), using
(218−29, 218, large)-Net in Base 3 — Upper bound on s
There is no (189, 218, large)-net in base 3, because
- 27 times m-reduction [i] would yield (189, 191, large)-net in base 3, but