Best Known (163, 163+27, s)-Nets in Base 3
(163, 163+27, 4545)-Net over F3 — Constructive and digital
Digital (163, 190, 4545)-net over F3, using
- 32 times duplication [i] based on digital (161, 188, 4545)-net over F3, using
- net defined by OOA [i] based on linear OOA(3188, 4545, F3, 27, 27) (dual of [(4545, 27), 122527, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3188, 59086, F3, 27) (dual of [59086, 58898, 28]-code), using
- 1 times truncation [i] based on linear OA(3189, 59087, F3, 28) (dual of [59087, 58898, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(22) [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(3151, 59049, F3, 23) (dual of [59049, 58898, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(38, 38, F3, 4) (dual of [38, 30, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(27) ⊂ Ce(22) [i] based on
- 1 times truncation [i] based on linear OA(3189, 59087, F3, 28) (dual of [59087, 58898, 29]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3188, 59086, F3, 27) (dual of [59086, 58898, 28]-code), using
- net defined by OOA [i] based on linear OOA(3188, 4545, F3, 27, 27) (dual of [(4545, 27), 122527, 28]-NRT-code), using
(163, 163+27, 26755)-Net over F3 — Digital
Digital (163, 190, 26755)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3190, 26755, F3, 2, 27) (dual of [(26755, 2), 53320, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3190, 29549, F3, 2, 27) (dual of [(29549, 2), 58908, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3190, 59098, F3, 27) (dual of [59098, 58908, 28]-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(39, 49, F3, 4) (dual of [49, 40, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- OOA 2-folding [i] based on linear OA(3190, 59098, F3, 27) (dual of [59098, 58908, 28]-code), using
- discarding factors / shortening the dual code based on linear OOA(3190, 29549, F3, 2, 27) (dual of [(29549, 2), 58908, 28]-NRT-code), using
(163, 163+27, large)-Net in Base 3 — Upper bound on s
There is no (163, 190, large)-net in base 3, because
- 25 times m-reduction [i] would yield (163, 165, large)-net in base 3, but