Best Known (15, 15+18, s)-Nets in Base 27
(15, 15+18, 112)-Net over F27 — Constructive and digital
Digital (15, 33, 112)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (2, 11, 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, 22, 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, 11, 48)-net over F27, using
(15, 15+18, 160)-Net in Base 27 — Constructive
(15, 33, 160)-net in base 27, using
- 7 times m-reduction [i] based on (15, 40, 160)-net in base 27, using
- base change [i] based on digital (5, 30, 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, 30, 160)-net over F81, using
(15, 15+18, 184)-Net over F27 — Digital
Digital (15, 33, 184)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2733, 184, F27, 18) (dual of [184, 151, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2733, 185, F27, 18) (dual of [185, 152, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(2733, 183, F27, 18) (dual of [183, 150, 19]-code), using an extension Ce(17) of the narrow-sense BCH-code C(I) with length 182 | 272−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2731, 183, F27, 17) (dual of [183, 152, 18]-code), using an extension Ce(16) of the narrow-sense BCH-code C(I) with length 182 | 272−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(2733, 185, F27, 18) (dual of [185, 152, 19]-code), using
(15, 15+18, 190)-Net in Base 27
(15, 33, 190)-net in base 27, using
- 3 times m-reduction [i] based on (15, 36, 190)-net in base 27, using
- base change [i] based on digital (6, 27, 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, 27, 190)-net over F81, using
(15, 15+18, 28251)-Net in Base 27 — Upper bound on s
There is no (15, 33, 28252)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 171807 095433 439946 946686 832522 596438 951962 081241 > 2733 [i]