Best Known (60−18, 60, s)-Nets in Base 3
(60−18, 60, 144)-Net over F3 — Constructive and digital
Digital (42, 60, 144)-net over F3, using
- trace code for nets [i] based on digital (2, 20, 48)-net over F27, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 2 and N(F) ≥ 48, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
(60−18, 60, 183)-Net over F3 — Digital
Digital (42, 60, 183)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(360, 183, F3, 18) (dual of [183, 123, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(360, 242, F3, 18) (dual of [242, 182, 19]-code), using
- the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(360, 242, F3, 18) (dual of [242, 182, 19]-code), using
(60−18, 60, 3135)-Net in Base 3 — Upper bound on s
There is no (42, 60, 3136)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 42409 128194 760631 937041 087617 > 360 [i]