Best Known (12, 27, s)-Nets in Base 27
(12, 27, 102)-Net over F27 — Constructive and digital
Digital (12, 27, 102)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (1, 8, 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 (4, 19, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 4 and N(F) ≥ 64, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- digital (1, 8, 38)-net over F27, using
(12, 27, 154)-Net over F27 — Digital
Digital (12, 27, 154)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2727, 154, F27, 15) (dual of [154, 127, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2727, 182, F27, 15) (dual of [182, 155, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 182 | 272−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(2727, 182, F27, 15) (dual of [182, 155, 16]-code), using
(12, 27, 160)-Net in Base 27 — Constructive
(12, 27, 160)-net in base 27, using
- 1 times m-reduction [i] based on (12, 28, 160)-net in base 27, using
- base change [i] based on digital (5, 21, 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
- base change [i] based on digital (5, 21, 160)-net over F81, using
(12, 27, 167)-Net in Base 27
(12, 27, 167)-net in base 27, using
- 1 times m-reduction [i] based on (12, 28, 167)-net in base 27, using
- base change [i] based on digital (5, 21, 167)-net over F81, using
- net from sequence [i] based on digital (5, 166)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 167, using
- net from sequence [i] based on digital (5, 166)-sequence over F81, using
- base change [i] based on digital (5, 21, 167)-net over F81, using
(12, 27, 26939)-Net in Base 27 — Upper bound on s
There is no (12, 27, 26940)-net in base 27, because
- 1 times m-reduction [i] would yield (12, 26, 26940)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 16 426117 427772 849985 909452 547533 209649 > 2726 [i]