Best Known (167, 167+29, s)-Nets in Base 3
(167, 167+29, 4219)-Net over F3 — Constructive and digital
Digital (167, 196, 4219)-net over F3, using
- 31 times duplication [i] based on digital (166, 195, 4219)-net over F3, using
- net defined by OOA [i] based on linear OOA(3195, 4219, F3, 29, 29) (dual of [(4219, 29), 122156, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3195, 59067, F3, 29) (dual of [59067, 58872, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3195, 59073, F3, 29) (dual of [59073, 58878, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- linear OA(3191, 59049, F3, 29) (dual of [59049, 58858, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3171, 59049, F3, 26) (dual of [59049, 58878, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(34, 24, F3, 2) (dual of [24, 20, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(3195, 59073, F3, 29) (dual of [59073, 58878, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3195, 59067, F3, 29) (dual of [59067, 58872, 30]-code), using
- net defined by OOA [i] based on linear OOA(3195, 4219, F3, 29, 29) (dual of [(4219, 29), 122156, 30]-NRT-code), using
(167, 167+29, 19691)-Net over F3 — Digital
Digital (167, 196, 19691)-net over F3, using
- 31 times duplication [i] based on digital (166, 195, 19691)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3195, 19691, F3, 3, 29) (dual of [(19691, 3), 58878, 30]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3195, 59073, F3, 29) (dual of [59073, 58878, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- linear OA(3191, 59049, F3, 29) (dual of [59049, 58858, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3171, 59049, F3, 26) (dual of [59049, 58878, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(34, 24, F3, 2) (dual of [24, 20, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- OOA 3-folding [i] based on linear OA(3195, 59073, F3, 29) (dual of [59073, 58878, 30]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3195, 19691, F3, 3, 29) (dual of [(19691, 3), 58878, 30]-NRT-code), using
(167, 167+29, large)-Net in Base 3 — Upper bound on s
There is no (167, 196, large)-net in base 3, because
- 27 times m-reduction [i] would yield (167, 169, large)-net in base 3, but