Best Known (147, 184, s)-Nets in Base 3
(147, 184, 688)-Net over F3 — Constructive and digital
Digital (147, 184, 688)-net over F3, using
- t-expansion [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
(147, 184, 2140)-Net over F3 — Digital
Digital (147, 184, 2140)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3184, 2140, F3, 37) (dual of [2140, 1956, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3184, 2231, F3, 37) (dual of [2231, 2047, 38]-code), using
- 5 times code embedding in larger space [i] based on linear OA(3179, 2226, F3, 37) (dual of [2226, 2047, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,15]) [i] based on
- 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(3141, 2188, F3, 31) (dual of [2188, 2047, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(310, 38, F3, 5) (dual of [38, 28, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(310, 39, F3, 5) (dual of [39, 29, 6]-code), using
- construction X applied to C([0,18]) ⊂ C([0,15]) [i] based on
- 5 times code embedding in larger space [i] based on linear OA(3179, 2226, F3, 37) (dual of [2226, 2047, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3184, 2231, F3, 37) (dual of [2231, 2047, 38]-code), using
(147, 184, 267796)-Net in Base 3 — Upper bound on s
There is no (147, 184, 267797)-net in base 3, because
- 1 times m-reduction [i] would yield (147, 183, 267797)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2056 896557 217264 386671 871005 915147 571579 005974 636104 640463 359417 223423 423403 240401 528881 > 3183 [i]