Best Known (72−12, 72, s)-Nets in Base 3
(72−12, 72, 3280)-Net over F3 — Constructive and digital
Digital (60, 72, 3280)-net over F3, using
- net defined by OOA [i] based on linear OOA(372, 3280, F3, 12, 12) (dual of [(3280, 12), 39288, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(372, 19680, F3, 12) (dual of [19680, 19608, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(372, 19683, F3, 12) (dual of [19683, 19611, 13]-code), using
- 1 times truncation [i] based on linear OA(373, 19684, F3, 13) (dual of [19684, 19611, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(373, 19684, F3, 13) (dual of [19684, 19611, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(372, 19683, F3, 12) (dual of [19683, 19611, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(372, 19680, F3, 12) (dual of [19680, 19608, 13]-code), using
(72−12, 72, 9841)-Net over F3 — Digital
Digital (60, 72, 9841)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(372, 9841, F3, 2, 12) (dual of [(9841, 2), 19610, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(372, 19682, F3, 12) (dual of [19682, 19610, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(372, 19683, F3, 12) (dual of [19683, 19611, 13]-code), using
- 1 times truncation [i] based on linear OA(373, 19684, F3, 13) (dual of [19684, 19611, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(373, 19684, F3, 13) (dual of [19684, 19611, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(372, 19683, F3, 12) (dual of [19683, 19611, 13]-code), using
- OOA 2-folding [i] based on linear OA(372, 19682, F3, 12) (dual of [19682, 19610, 13]-code), using
(72−12, 72, 795507)-Net in Base 3 — Upper bound on s
There is no (60, 72, 795508)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 22528 527271 730987 331681 892208 557945 > 372 [i]