Best Known (42−22, 42, s)-Nets in Base 27
(42−22, 42, 132)-Net over F27 — Constructive and digital
Digital (20, 42, 132)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (4, 15, 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 (5, 27, 68)-net over F27, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 5 and N(F) ≥ 68, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
- digital (4, 15, 64)-net over F27, using
(42−22, 42, 172)-Net in Base 27 — Constructive
(20, 42, 172)-net in base 27, using
- 10 times m-reduction [i] based on (20, 52, 172)-net in base 27, using
- base change [i] based on digital (7, 39, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- base change [i] based on digital (7, 39, 172)-net over F81, using
(42−22, 42, 266)-Net over F27 — Digital
Digital (20, 42, 266)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2742, 266, F27, 22) (dual of [266, 224, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(2742, 364, F27, 22) (dual of [364, 322, 23]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 364 | 272−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(2742, 364, F27, 22) (dual of [364, 322, 23]-code), using
(42−22, 42, 55106)-Net in Base 27 — Upper bound on s
There is no (20, 42, 55107)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 1 310073 496402 248646 450271 935898 396508 438445 610390 486515 812019 > 2742 [i]