Best Known (90, 90+18, s)-Nets in Base 3
(90, 90+18, 2187)-Net over F3 — Constructive and digital
Digital (90, 108, 2187)-net over F3, using
- net defined by OOA [i] based on linear OOA(3108, 2187, F3, 18, 18) (dual of [(2187, 18), 39258, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3108, 19683, F3, 18) (dual of [19683, 19575, 19]-code), using
- 1 times truncation [i] based on linear OA(3109, 19684, F3, 19) (dual of [19684, 19575, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(3109, 19684, F3, 19) (dual of [19684, 19575, 20]-code), using
- OA 9-folding and stacking [i] based on linear OA(3108, 19683, F3, 18) (dual of [19683, 19575, 19]-code), using
(90, 90+18, 7544)-Net over F3 — Digital
Digital (90, 108, 7544)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3108, 7544, F3, 2, 18) (dual of [(7544, 2), 14980, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3108, 9841, F3, 2, 18) (dual of [(9841, 2), 19574, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3108, 19682, F3, 18) (dual of [19682, 19574, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3108, 19683, F3, 18) (dual of [19683, 19575, 19]-code), using
- 1 times truncation [i] based on linear OA(3109, 19684, F3, 19) (dual of [19684, 19575, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(3109, 19684, F3, 19) (dual of [19684, 19575, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3108, 19683, F3, 18) (dual of [19683, 19575, 19]-code), using
- OOA 2-folding [i] based on linear OA(3108, 19682, F3, 18) (dual of [19682, 19574, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(3108, 9841, F3, 2, 18) (dual of [(9841, 2), 19574, 19]-NRT-code), using
(90, 90+18, 1101978)-Net in Base 3 — Upper bound on s
There is no (90, 108, 1101979)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 3381 403069 223157 927501 409056 985879 981465 060769 618919 > 3108 [i]