Best Known (182−39, 182, s)-Nets in Base 3
(182−39, 182, 640)-Net over F3 — Constructive and digital
Digital (143, 182, 640)-net over F3, using
- 2 times m-reduction [i] based on digital (143, 184, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 46, 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, 46, 160)-net over F81, using
(182−39, 182, 1547)-Net over F3 — Digital
Digital (143, 182, 1547)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3182, 1547, F3, 39) (dual of [1547, 1365, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3182, 2186, F3, 39) (dual of [2186, 2004, 40]-code), using
- 1 times truncation [i] based on linear OA(3183, 2187, F3, 40) (dual of [2187, 2004, 41]-code), using
- an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- 1 times truncation [i] based on linear OA(3183, 2187, F3, 40) (dual of [2187, 2004, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3182, 2186, F3, 39) (dual of [2186, 2004, 40]-code), using
(182−39, 182, 139102)-Net in Base 3 — Upper bound on s
There is no (143, 182, 139103)-net in base 3, because
- 1 times m-reduction [i] would yield (143, 181, 139103)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 228 541995 426278 889922 518137 618442 379207 181062 236214 986407 405006 530726 922861 157642 702683 > 3181 [i]