Best Known (161, 188, s)-Nets in Base 3
(161, 188, 4545)-Net over F3 — Constructive and digital
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
(161, 188, 24413)-Net over F3 — Digital
Digital (161, 188, 24413)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3188, 24413, F3, 2, 27) (dual of [(24413, 2), 48638, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3188, 29543, F3, 2, 27) (dual of [(29543, 2), 58898, 28]-NRT-code), using
- OOA 2-folding [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 2-folding [i] based on linear OA(3188, 59086, F3, 27) (dual of [59086, 58898, 28]-code), using
- discarding factors / shortening the dual code based on linear OOA(3188, 29543, F3, 2, 27) (dual of [(29543, 2), 58898, 28]-NRT-code), using
(161, 188, large)-Net in Base 3 — Upper bound on s
There is no (161, 188, large)-net in base 3, because
- 25 times m-reduction [i] would yield (161, 163, large)-net in base 3, but