Best Known (3, 3+6, s)-Nets in Base 27
(3, 3+6, 56)-Net over F27 — Constructive and digital
Digital (3, 9, 56)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 0 and N(F) ≥ 28, using
- the rational function field F27(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- digital (0, 6, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 3, 28)-net over F27, using
(3, 3+6, 58)-Net over F27 — Digital
Digital (3, 9, 58)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(279, 58, F27, 6) (dual of [58, 49, 7]-code), using
- construction X applied to C([25,30]) ⊂ C([26,30]) [i] based on
- linear OA(279, 56, F27, 6) (dual of [56, 47, 7]-code), using the BCH-code C(I) with length 56 | 272−1, defining interval I = {25,26,…,30}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(277, 56, F27, 5) (dual of [56, 49, 6]-code), using the BCH-code C(I) with length 56 | 272−1, defining interval I = {26,27,28,29,30}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to C([25,30]) ⊂ C([26,30]) [i] based on
(3, 3+6, 82)-Net in Base 27 — Constructive
(3, 9, 82)-net in base 27, using
- 3 times m-reduction [i] based on (3, 12, 82)-net in base 27, using
- base change [i] based on digital (0, 9, 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
- base change [i] based on digital (0, 9, 82)-net over F81, using
(3, 3+6, 1374)-Net in Base 27 — Upper bound on s
There is no (3, 9, 1375)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 7 633663 859251 > 279 [i]