Best Known (31−17, 31, s)-Nets in Base 27
(31−17, 31, 112)-Net over F27 — Constructive and digital
Digital (14, 31, 112)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (2, 10, 48)-net over F27, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 2 and N(F) ≥ 48, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- digital (4, 21, 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 (2, 10, 48)-net over F27, using
(31−17, 31, 160)-Net in Base 27 — Constructive
(14, 31, 160)-net in base 27, using
- 5 times m-reduction [i] based on (14, 36, 160)-net in base 27, using
- base change [i] based on digital (5, 27, 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, 27, 160)-net over F81, using
(31−17, 31, 174)-Net over F27 — Digital
Digital (14, 31, 174)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2731, 174, F27, 17) (dual of [174, 143, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2731, 182, F27, 17) (dual of [182, 151, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 182 | 272−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(2731, 182, F27, 17) (dual of [182, 151, 18]-code), using
(31−17, 31, 190)-Net in Base 27
(14, 31, 190)-net in base 27, using
- 1 times m-reduction [i] based on (14, 32, 190)-net in base 27, using
- base change [i] based on digital (6, 24, 190)-net over F81, using
- net from sequence [i] based on digital (6, 189)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 6 and N(F) ≥ 190, using
- net from sequence [i] based on digital (6, 189)-sequence over F81, using
- base change [i] based on digital (6, 24, 190)-net over F81, using
(31−17, 31, 33750)-Net in Base 27 — Upper bound on s
There is no (14, 31, 33751)-net in base 27, because
- 1 times m-reduction [i] would yield (14, 30, 33751)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 8 728735 899592 274599 415938 414966 488291 650481 > 2730 [i]