Best Known (190−28, 190, s)-Nets in Base 3
(190−28, 190, 4220)-Net over F3 — Constructive and digital
Digital (162, 190, 4220)-net over F3, using
- 31 times duplication [i] based on digital (161, 189, 4220)-net over F3, using
- net defined by OOA [i] based on linear OOA(3189, 4220, F3, 28, 28) (dual of [(4220, 28), 117971, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(3189, 59080, F3, 28) (dual of [59080, 58891, 29]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(3189, 59087, F3, 28) (dual of [59087, 58898, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(3189, 59080, F3, 28) (dual of [59080, 58891, 29]-code), using
- net defined by OOA [i] based on linear OOA(3189, 4220, F3, 28, 28) (dual of [(4220, 28), 117971, 29]-NRT-code), using
(190−28, 190, 19696)-Net over F3 — Digital
Digital (162, 190, 19696)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3190, 19696, F3, 3, 28) (dual of [(19696, 3), 58898, 29]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3190, 59088, F3, 28) (dual of [59088, 58898, 29]-code), using
- 1 times code embedding in larger space [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 code embedding in larger space [i] based on linear OA(3189, 59087, F3, 28) (dual of [59087, 58898, 29]-code), using
- OOA 3-folding [i] based on linear OA(3190, 59088, F3, 28) (dual of [59088, 58898, 29]-code), using
(190−28, 190, large)-Net in Base 3 — Upper bound on s
There is no (162, 190, large)-net in base 3, because
- 26 times m-reduction [i] would yield (162, 164, large)-net in base 3, but