Best Known (141−23, 141, s)-Nets in Base 3
(141−23, 141, 1791)-Net over F3 — Constructive and digital
Digital (118, 141, 1791)-net over F3, using
- 31 times duplication [i] based on digital (117, 140, 1791)-net over F3, using
- net defined by OOA [i] based on linear OOA(3140, 1791, F3, 23, 23) (dual of [(1791, 23), 41053, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3140, 19702, F3, 23) (dual of [19702, 19562, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3140, 19705, F3, 23) (dual of [19705, 19565, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- linear OA(3136, 19683, F3, 23) (dual of [19683, 19547, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3118, 19683, F3, 20) (dual of [19683, 19565, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(34, 22, F3, 2) (dual of [22, 18, 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(22) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3140, 19705, F3, 23) (dual of [19705, 19565, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3140, 19702, F3, 23) (dual of [19702, 19562, 24]-code), using
- net defined by OOA [i] based on linear OOA(3140, 1791, F3, 23, 23) (dual of [(1791, 23), 41053, 24]-NRT-code), using
(141−23, 141, 8576)-Net over F3 — Digital
Digital (118, 141, 8576)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3141, 8576, F3, 2, 23) (dual of [(8576, 2), 17011, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3141, 9853, F3, 2, 23) (dual of [(9853, 2), 19565, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3141, 19706, F3, 23) (dual of [19706, 19565, 24]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3140, 19705, F3, 23) (dual of [19705, 19565, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- linear OA(3136, 19683, F3, 23) (dual of [19683, 19547, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3118, 19683, F3, 20) (dual of [19683, 19565, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(34, 22, F3, 2) (dual of [22, 18, 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(22) ⊂ Ce(19) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3140, 19705, F3, 23) (dual of [19705, 19565, 24]-code), using
- OOA 2-folding [i] based on linear OA(3141, 19706, F3, 23) (dual of [19706, 19565, 24]-code), using
- discarding factors / shortening the dual code based on linear OOA(3141, 9853, F3, 2, 23) (dual of [(9853, 2), 19565, 24]-NRT-code), using
(141−23, 141, 2900246)-Net in Base 3 — Upper bound on s
There is no (118, 141, 2900247)-net in base 3, because
- 1 times m-reduction [i] would yield (118, 140, 2900247)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 6 265797 833209 257591 825725 855949 591721 124825 959655 858904 330976 899275 > 3140 [i]