Best Known (166−25, 166, s)-Nets in Base 3
(166−25, 166, 4922)-Net over F3 — Constructive and digital
Digital (141, 166, 4922)-net over F3, using
- 31 times duplication [i] based on digital (140, 165, 4922)-net over F3, using
- net defined by OOA [i] based on linear OOA(3165, 4922, F3, 25, 25) (dual of [(4922, 25), 122885, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3165, 59065, F3, 25) (dual of [59065, 58900, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3165, 59073, F3, 25) (dual of [59073, 58908, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(3161, 59049, F3, 25) (dual of [59049, 58888, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3141, 59049, F3, 22) (dual of [59049, 58908, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(24) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3165, 59073, F3, 25) (dual of [59073, 58908, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3165, 59065, F3, 25) (dual of [59065, 58900, 26]-code), using
- net defined by OOA [i] based on linear OOA(3165, 4922, F3, 25, 25) (dual of [(4922, 25), 122885, 26]-NRT-code), using
(166−25, 166, 19691)-Net over F3 — Digital
Digital (141, 166, 19691)-net over F3, using
- 31 times duplication [i] based on digital (140, 165, 19691)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3165, 19691, F3, 3, 25) (dual of [(19691, 3), 58908, 26]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3165, 59073, F3, 25) (dual of [59073, 58908, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(3161, 59049, F3, 25) (dual of [59049, 58888, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3141, 59049, F3, 22) (dual of [59049, 58908, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(24) ⊂ Ce(21) [i] based on
- OOA 3-folding [i] based on linear OA(3165, 59073, F3, 25) (dual of [59073, 58908, 26]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3165, 19691, F3, 3, 25) (dual of [(19691, 3), 58908, 26]-NRT-code), using
(166−25, 166, large)-Net in Base 3 — Upper bound on s
There is no (141, 166, large)-net in base 3, because
- 23 times m-reduction [i] would yield (141, 143, large)-net in base 3, but