Best Known (66−19, 66, s)-Nets in Base 3
(66−19, 66, 156)-Net over F3 — Constructive and digital
Digital (47, 66, 156)-net over F3, using
- trace code for nets [i] based on digital (3, 22, 52)-net over F27, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 3 and N(F) ≥ 52, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
(66−19, 66, 226)-Net over F3 — Digital
Digital (47, 66, 226)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(366, 226, F3, 19) (dual of [226, 160, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(366, 259, F3, 19) (dual of [259, 193, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- linear OA(361, 244, F3, 19) (dual of [244, 183, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 244 | 310−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(351, 244, F3, 15) (dual of [244, 193, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 244 | 310−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(35, 15, F3, 3) (dual of [15, 10, 4]-code or 15-cap in PG(4,3)), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(366, 259, F3, 19) (dual of [259, 193, 20]-code), using
(66−19, 66, 5780)-Net in Base 3 — Upper bound on s
There is no (47, 66, 5781)-net in base 3, because
- 1 times m-reduction [i] would yield (47, 65, 5781)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 10 310338 355034 173215 323232 084875 > 365 [i]