Best Known (165, 165+28, s)-Nets in Base 3
(165, 165+28, 4221)-Net over F3 — Constructive and digital
Digital (165, 193, 4221)-net over F3, using
- 31 times duplication [i] based on digital (164, 192, 4221)-net over F3, using
- net defined by OOA [i] based on linear OOA(3192, 4221, F3, 28, 28) (dual of [(4221, 28), 117996, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(3192, 59094, F3, 28) (dual of [59094, 58902, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3192, 59100, F3, 28) (dual of [59100, 58908, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- linear OA(3181, 59049, F3, 28) (dual of [59049, 58868, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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(311, 51, F3, 5) (dual of [51, 40, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3192, 59100, F3, 28) (dual of [59100, 58908, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(3192, 59094, F3, 28) (dual of [59094, 58902, 29]-code), using
- net defined by OOA [i] based on linear OOA(3192, 4221, F3, 28, 28) (dual of [(4221, 28), 117996, 29]-NRT-code), using
(165, 165+28, 22456)-Net over F3 — Digital
Digital (165, 193, 22456)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3193, 22456, F3, 2, 28) (dual of [(22456, 2), 44719, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3193, 29550, F3, 2, 28) (dual of [(29550, 2), 58907, 29]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3192, 29550, F3, 2, 28) (dual of [(29550, 2), 58908, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3192, 59100, F3, 28) (dual of [59100, 58908, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- linear OA(3181, 59049, F3, 28) (dual of [59049, 58868, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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(311, 51, F3, 5) (dual of [51, 40, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- OOA 2-folding [i] based on linear OA(3192, 59100, F3, 28) (dual of [59100, 58908, 29]-code), using
- 31 times duplication [i] based on linear OOA(3192, 29550, F3, 2, 28) (dual of [(29550, 2), 58908, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3193, 29550, F3, 2, 28) (dual of [(29550, 2), 58907, 29]-NRT-code), using
(165, 165+28, large)-Net in Base 3 — Upper bound on s
There is no (165, 193, large)-net in base 3, because
- 26 times m-reduction [i] would yield (165, 167, large)-net in base 3, but