Best Known (66, 66+19, s)-Nets in Base 3
(66, 66+19, 464)-Net over F3 — Constructive and digital
Digital (66, 85, 464)-net over F3, using
- 31 times duplication [i] based on digital (65, 84, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 21, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- trace code for nets [i] based on digital (2, 21, 116)-net over F81, using
(66, 66+19, 1000)-Net over F3 — Digital
Digital (66, 85, 1000)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(385, 1000, F3, 2, 19) (dual of [(1000, 2), 1915, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(385, 1094, F3, 2, 19) (dual of [(1094, 2), 2103, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(385, 2188, F3, 19) (dual of [2188, 2103, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- OOA 2-folding [i] based on linear OA(385, 2188, F3, 19) (dual of [2188, 2103, 20]-code), using
- discarding factors / shortening the dual code based on linear OOA(385, 1094, F3, 2, 19) (dual of [(1094, 2), 2103, 20]-NRT-code), using
(66, 66+19, 58855)-Net in Base 3 — Upper bound on s
There is no (66, 85, 58856)-net in base 3, because
- 1 times m-reduction [i] would yield (66, 84, 58856)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 11972 587751 018162 588840 871035 871478 714705 > 384 [i]