Best Known (149, 149+41, s)-Nets in Base 3
(149, 149+41, 640)-Net over F3 — Constructive and digital
Digital (149, 190, 640)-net over F3, using
- 2 times m-reduction [i] based on digital (149, 192, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 48, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 48, 160)-net over F81, using
(149, 149+41, 1544)-Net over F3 — Digital
Digital (149, 190, 1544)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3190, 1544, F3, 41) (dual of [1544, 1354, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3190, 2187, F3, 41) (dual of [2187, 1997, 42]-code), using
- an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- discarding factors / shortening the dual code based on linear OA(3190, 2187, F3, 41) (dual of [2187, 1997, 42]-code), using
(149, 149+41, 133970)-Net in Base 3 — Upper bound on s
There is no (149, 190, 133971)-net in base 3, because
- 1 times m-reduction [i] would yield (149, 189, 133971)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1 499530 743774 528563 154865 740746 210899 016751 797317 673982 701940 824779 247943 926409 910573 558201 > 3189 [i]