Best Known (93−18, 93, s)-Nets in Base 3
(93−18, 93, 640)-Net over F3 — Constructive and digital
Digital (75, 93, 640)-net over F3, using
- 31 times duplication [i] based on digital (74, 92, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 23, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 23, 160)-net over F81, using
(93−18, 93, 1869)-Net over F3 — Digital
Digital (75, 93, 1869)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(393, 1869, F3, 18) (dual of [1869, 1776, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(393, 2224, F3, 18) (dual of [2224, 2131, 19]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [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]
- linear OA(357, 2188, F3, 13) (dual of [2188, 2131, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(38, 36, F3, 4) (dual of [36, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(393, 2224, F3, 18) (dual of [2224, 2131, 19]-code), using
(93−18, 93, 176584)-Net in Base 3 — Upper bound on s
There is no (75, 93, 176585)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 235 656461 205775 652844 742258 663973 967498 899955 > 393 [i]