Best Known (18, 18+6, s)-Nets in Base 3
(18, 18+6, 328)-Net over F3 — Constructive and digital
Digital (18, 24, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 6, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
(18, 18+6, 610)-Net over F3 — Digital
Digital (18, 24, 610)-net over F3, using
- net defined by OOA [i] based on linear OOA(324, 610, F3, 6, 6) (dual of [(610, 6), 3636, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(324, 610, F3, 5, 6) (dual of [(610, 5), 3026, 7]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(324, 610, F3, 6) (dual of [610, 586, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(324, 728, F3, 6) (dual of [728, 704, 7]-code), using
- the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(324, 728, F3, 6) (dual of [728, 704, 7]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(324, 610, F3, 6) (dual of [610, 586, 7]-code), using
- appending kth column [i] based on linear OOA(324, 610, F3, 5, 6) (dual of [(610, 5), 3026, 7]-NRT-code), using
(18, 18+6, 5958)-Net in Base 3 — Upper bound on s
There is no (18, 24, 5959)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 282491 436315 > 324 [i]