Best Known (178, 224, s)-Nets in Base 3
(178, 224, 688)-Net over F3 — Constructive and digital
Digital (178, 224, 688)-net over F3, using
- 4 times m-reduction [i] based on digital (178, 228, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 57, 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, 57, 172)-net over F81, using
(178, 224, 2219)-Net over F3 — Digital
Digital (178, 224, 2219)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3224, 2219, F3, 46) (dual of [2219, 1995, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(3224, 2228, F3, 46) (dual of [2228, 2004, 47]-code), using
- 3 times code embedding in larger space [i] based on linear OA(3221, 2225, F3, 46) (dual of [2225, 2004, 47]-code), using
- construction X applied to Ce(45) ⊂ Ce(39) [i] based on
- linear OA(3211, 2187, F3, 46) (dual of [2187, 1976, 47]-code), using an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- 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]
- 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 Ce(45) ⊂ Ce(39) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(3221, 2225, F3, 46) (dual of [2225, 2004, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(3224, 2228, F3, 46) (dual of [2228, 2004, 47]-code), using
(178, 224, 208990)-Net in Base 3 — Upper bound on s
There is no (178, 224, 208991)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 75024 859528 714767 863667 417060 256900 296233 413658 494148 188508 255919 600287 921847 734117 056061 074510 394337 296211 > 3224 [i]