Best Known (37−18, 37, s)-Nets in Base 27
(37−18, 37, 140)-Net over F27 — Constructive and digital
Digital (19, 37, 140)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (4, 13, 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 (6, 24, 76)-net over F27, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 6 and N(F) ≥ 76, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- digital (4, 13, 64)-net over F27, using
(37−18, 37, 172)-Net in Base 27 — Constructive
(19, 37, 172)-net in base 27, using
- 11 times m-reduction [i] based on (19, 48, 172)-net in base 27, using
- base change [i] based on digital (7, 36, 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, 36, 172)-net over F81, using
(37−18, 37, 428)-Net over F27 — Digital
Digital (19, 37, 428)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2737, 428, F27, 18) (dual of [428, 391, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2737, 737, F27, 18) (dual of [737, 700, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- linear OA(2735, 729, F27, 18) (dual of [729, 694, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2729, 729, F27, 15) (dual of [729, 700, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(272, 8, F27, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,27)), using
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- Reed–Solomon code RS(25,27) [i]
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(2737, 737, F27, 18) (dual of [737, 700, 19]-code), using
(37−18, 37, 122252)-Net in Base 27 — Upper bound on s
There is no (19, 37, 122253)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 91300 209978 298806 235143 859068 136623 244412 468193 103603 > 2737 [i]