Best Known (145, 145+38, s)-Nets in Base 3
(145, 145+38, 688)-Net over F3 — Constructive and digital
Digital (145, 183, 688)-net over F3, using
- 1 times m-reduction [i] based on digital (145, 184, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 46, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 46, 172)-net over F81, using
(145, 145+38, 1811)-Net over F3 — Digital
Digital (145, 183, 1811)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3183, 1811, F3, 38) (dual of [1811, 1628, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3183, 2215, F3, 38) (dual of [2215, 2032, 39]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3182, 2214, F3, 38) (dual of [2214, 2032, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(33) [i] based on
- linear OA(3176, 2187, F3, 38) (dual of [2187, 2011, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3155, 2187, F3, 34) (dual of [2187, 2032, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(36, 27, F3, 3) (dual of [27, 21, 4]-code or 27-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(37) ⊂ Ce(33) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3182, 2214, F3, 38) (dual of [2214, 2032, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3183, 2215, F3, 38) (dual of [2215, 2032, 39]-code), using
(145, 145+38, 156158)-Net in Base 3 — Upper bound on s
There is no (145, 183, 156159)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 2056 961265 279347 892811 014189 942889 864479 078297 988048 229515 710212 884975 335560 641355 868123 > 3183 [i]