Best Known (4, 4+6, s)-Nets in Base 27
(4, 4+6, 66)-Net over F27 — Constructive and digital
Digital (4, 10, 66)-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 (1, 7, 38)-net over F27, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 1 and N(F) ≥ 38, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- digital (0, 3, 28)-net over F27, using
(4, 4+6, 100)-Net in Base 27 — Constructive
(4, 10, 100)-net in base 27, using
- 2 times m-reduction [i] based on (4, 12, 100)-net in base 27, using
- base change [i] based on digital (1, 9, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- base change [i] based on digital (1, 9, 100)-net over F81, using
(4, 4+6, 107)-Net over F27 — Digital
Digital (4, 10, 107)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2710, 107, F27, 6) (dual of [107, 97, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(2710, 105, F27, 6) (dual of [105, 95, 7]-code), using an extension Ce(5) of the narrow-sense BCH-code C(I) with length 104 | 272−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(278, 105, F27, 5) (dual of [105, 97, 6]-code), using an extension Ce(4) of the narrow-sense BCH-code C(I) with length 104 | 272−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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 Ce(5) ⊂ Ce(4) [i] based on
(4, 4+6, 4125)-Net in Base 27 — Upper bound on s
There is no (4, 10, 4126)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 205 924770 964433 > 2710 [i]