Best Known (178, 178+47, s)-Nets in Base 3
(178, 178+47, 688)-Net over F3 — Constructive and digital
Digital (178, 225, 688)-net over F3, using
- 3 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, 178+47, 2048)-Net over F3 — Digital
Digital (178, 225, 2048)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3225, 2048, F3, 47) (dual of [2048, 1823, 48]-code), using
- discarding factors / shortening the dual code based on linear OA(3225, 2215, F3, 47) (dual of [2215, 1990, 48]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3224, 2214, F3, 47) (dual of [2214, 1990, 48]-code), using
- construction X applied to Ce(46) ⊂ Ce(42) [i] based on
- linear OA(3218, 2187, F3, 47) (dual of [2187, 1969, 48]-code), using an extension Ce(46) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,46], and designed minimum distance d ≥ |I|+1 = 47 [i]
- linear OA(3197, 2187, F3, 43) (dual of [2187, 1990, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [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(46) ⊂ Ce(42) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3224, 2214, F3, 47) (dual of [2214, 1990, 48]-code), using
- discarding factors / shortening the dual code based on linear OA(3225, 2215, F3, 47) (dual of [2215, 1990, 48]-code), using
(178, 178+47, 208990)-Net in Base 3 — Upper bound on s
There is no (178, 225, 208991)-net in base 3, because
- 1 times m-reduction [i] would yield (178, 224, 208991)-net in base 3, but
- 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]