Best Known (223−46, 223, s)-Nets in Base 3
(223−46, 223, 688)-Net over F3 — Constructive and digital
Digital (177, 223, 688)-net over F3, using
- t-expansion [i] based on digital (175, 223, 688)-net over F3, using
- 1 times m-reduction [i] based on digital (175, 224, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 56, 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, 56, 172)-net over F81, using
- 1 times m-reduction [i] based on digital (175, 224, 688)-net over F3, using
(223−46, 223, 2163)-Net over F3 — Digital
Digital (177, 223, 2163)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3223, 2163, F3, 46) (dual of [2163, 1940, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(3223, 2227, F3, 46) (dual of [2227, 2004, 47]-code), using
- 2 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
- 2 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(3223, 2227, F3, 46) (dual of [2227, 2004, 47]-code), using
(223−46, 223, 199241)-Net in Base 3 — Upper bound on s
There is no (177, 223, 199242)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 25008 284172 037573 012715 094983 462077 082307 767972 762757 050837 147508 171622 089927 623435 099188 124848 109435 604393 > 3223 [i]