Best Known (145, 145+39, s)-Nets in Base 3
(145, 145+39, 688)-Net over F3 — Constructive and digital
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
(145, 145+39, 1644)-Net over F3 — Digital
Digital (145, 184, 1644)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3184, 1644, F3, 39) (dual of [1644, 1460, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3184, 2203, F3, 39) (dual of [2203, 2019, 40]-code), using
- construction X applied to C([0,19]) ⊂ C([0,18]) [i] based on
- linear OA(3183, 2188, F3, 39) (dual of [2188, 2005, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- linear OA(3169, 2188, F3, 37) (dual of [2188, 2019, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(31, 15, F3, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,19]) ⊂ C([0,18]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3184, 2203, F3, 39) (dual of [2203, 2019, 40]-code), using
(145, 145+39, 156158)-Net in Base 3 — Upper bound on s
There is no (145, 184, 156159)-net in base 3, because
- 1 times m-reduction [i] would yield (145, 183, 156159)-net in base 3, but
- 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]