Best Known (157, 181, s)-Nets in Base 3
(157, 181, 14764)-Net over F3 — Constructive and digital
Digital (157, 181, 14764)-net over F3, using
- t-expansion [i] based on digital (156, 181, 14764)-net over F3, using
- net defined by OOA [i] based on linear OOA(3181, 14764, F3, 25, 25) (dual of [(14764, 25), 368919, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3181, 177169, F3, 25) (dual of [177169, 176988, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3181, 177173, F3, 25) (dual of [177173, 176992, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(3177, 177147, F3, 25) (dual of [177147, 176970, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3155, 177147, F3, 22) (dual of [177147, 176992, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(34, 26, F3, 2) (dual of [26, 22, 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(24) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3181, 177173, F3, 25) (dual of [177173, 176992, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3181, 177169, F3, 25) (dual of [177169, 176988, 26]-code), using
- net defined by OOA [i] based on linear OOA(3181, 14764, F3, 25, 25) (dual of [(14764, 25), 368919, 26]-NRT-code), using
(157, 181, 59058)-Net over F3 — Digital
Digital (157, 181, 59058)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3181, 59058, F3, 3, 24) (dual of [(59058, 3), 176993, 25]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3181, 177174, F3, 24) (dual of [177174, 176993, 25]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(3177, 177148, F3, 25) (dual of [177148, 176971, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(3155, 177148, F3, 21) (dual of [177148, 176993, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(34, 26, F3, 2) (dual of [26, 22, 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 C([0,12]) ⊂ C([0,10]) [i] based on
- OOA 3-folding [i] based on linear OA(3181, 177174, F3, 24) (dual of [177174, 176993, 25]-code), using
(157, 181, large)-Net in Base 3 — Upper bound on s
There is no (157, 181, large)-net in base 3, because
- 22 times m-reduction [i] would yield (157, 159, large)-net in base 3, but