Best Known (174, 174+47, s)-Nets in Base 3
(174, 174+47, 688)-Net over F3 — Constructive and digital
Digital (174, 221, 688)-net over F3, using
- 31 times duplication [i] based on digital (173, 220, 688)-net over F3, using
- t-expansion [i] based on digital (172, 220, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 55, 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, 55, 172)-net over F81, using
- t-expansion [i] based on digital (172, 220, 688)-net over F3, using
(174, 174+47, 1854)-Net over F3 — Digital
Digital (174, 221, 1854)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3221, 1854, F3, 47) (dual of [1854, 1633, 48]-code), using
- discarding factors / shortening the dual code based on linear OA(3221, 2200, F3, 47) (dual of [2200, 1979, 48]-code), using
- construction X applied to Ce(46) ⊂ Ce(43) [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(3204, 2187, F3, 44) (dual of [2187, 1983, 45]-code), using an extension Ce(43) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- construction X applied to Ce(46) ⊂ Ce(43) [i] based on
- discarding factors / shortening the dual code based on linear OA(3221, 2200, F3, 47) (dual of [2200, 1979, 48]-code), using
(174, 174+47, 172638)-Net in Base 3 — Upper bound on s
There is no (174, 221, 172639)-net in base 3, because
- 1 times m-reduction [i] would yield (174, 220, 172639)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 926 173008 403253 512376 152540 974447 852421 183606 746830 472609 088636 931367 209418 437547 729558 482740 178989 672275 > 3220 [i]