Best Known (109−19, 109, s)-Nets in Base 3
(109−19, 109, 2187)-Net over F3 — Constructive and digital
Digital (90, 109, 2187)-net over F3, using
- net defined by OOA [i] based on linear OOA(3109, 2187, F3, 19, 19) (dual of [(2187, 19), 41444, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [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]
- OOA 9-folding and stacking with additional row [i] based on linear OA(3109, 19684, F3, 19) (dual of [19684, 19575, 20]-code), using
(109−19, 109, 6561)-Net over F3 — Digital
Digital (90, 109, 6561)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3109, 6561, F3, 3, 19) (dual of [(6561, 3), 19574, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3109, 19683, F3, 19) (dual of [19683, 19574, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OOA 3-folding [i] based on linear OA(3109, 19683, F3, 19) (dual of [19683, 19574, 20]-code), using
(109−19, 109, 1101978)-Net in Base 3 — Upper bound on s
There is no (90, 109, 1101979)-net in base 3, because
- 1 times m-reduction [i] would yield (90, 108, 1101979)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 3381 403069 223157 927501 409056 985879 981465 060769 618919 > 3108 [i]