Best Known (174, 174+37, s)-Nets in Base 3
(174, 174+37, 896)-Net over F3 — Constructive and digital
Digital (174, 211, 896)-net over F3, using
- t-expansion [i] based on digital (172, 211, 896)-net over F3, using
- 1 times m-reduction [i] based on digital (172, 212, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 53, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 53, 224)-net over F81, using
- 1 times m-reduction [i] based on digital (172, 212, 896)-net over F3, using
(174, 174+37, 5036)-Net over F3 — Digital
Digital (174, 211, 5036)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3211, 5036, F3, 37) (dual of [5036, 4825, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3211, 6581, F3, 37) (dual of [6581, 6370, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([1,18]) [i] based on
- linear OA(3193, 6562, F3, 37) (dual of [6562, 6369, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(3192, 6562, F3, 18) (dual of [6562, 6370, 19]-code), using the narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(318, 19, F3, 18) (dual of [19, 1, 19]-code or 19-arc in PG(17,3)), using
- dual of repetition code with length 19 [i]
- construction X applied to C([0,18]) ⊂ C([1,18]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3211, 6581, F3, 37) (dual of [6581, 6370, 38]-code), using
(174, 174+37, 1391583)-Net in Base 3 — Upper bound on s
There is no (174, 211, 1391584)-net in base 3, because
- 1 times m-reduction [i] would yield (174, 210, 1391584)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 15684 363132 858664 679423 943635 945519 857279 403980 973766 679881 285985 327508 208283 519670 696060 206759 678785 > 3210 [i]