Best Known (41, 41+40, s)-Nets in Base 27
(41, 41+40, 190)-Net over F27 — Constructive and digital
Digital (41, 81, 190)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (10, 30, 94)-net over F27, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 10 and N(F) ≥ 94, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- digital (11, 51, 96)-net over F27, using
- net from sequence [i] based on digital (11, 95)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 11 and N(F) ≥ 96, using
- net from sequence [i] based on digital (11, 95)-sequence over F27, using
- digital (10, 30, 94)-net over F27, using
(41, 41+40, 370)-Net in Base 27 — Constructive
(41, 81, 370)-net in base 27, using
- 19 times m-reduction [i] based on (41, 100, 370)-net in base 27, using
- base change [i] based on digital (16, 75, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- base change [i] based on digital (16, 75, 370)-net over F81, using
(41, 41+40, 579)-Net over F27 — Digital
Digital (41, 81, 579)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2781, 579, F27, 40) (dual of [579, 498, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(2781, 746, F27, 40) (dual of [746, 665, 41]-code), using
- construction X applied to Ce(39) ⊂ Ce(33) [i] based on
- linear OA(2776, 729, F27, 40) (dual of [729, 653, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(2764, 729, F27, 34) (dual of [729, 665, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(275, 17, F27, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,27)), using
- discarding factors / shortening the dual code based on linear OA(275, 27, F27, 5) (dual of [27, 22, 6]-code or 27-arc in PG(4,27)), using
- Reed–Solomon code RS(22,27) [i]
- discarding factors / shortening the dual code based on linear OA(275, 27, F27, 5) (dual of [27, 22, 6]-code or 27-arc in PG(4,27)), using
- construction X applied to Ce(39) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(2781, 746, F27, 40) (dual of [746, 665, 41]-code), using
(41, 41+40, 200139)-Net in Base 27 — Upper bound on s
There is no (41, 81, 200140)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 87 189845 858913 409855 848504 231729 014995 986997 563447 798596 356194 186102 946118 727105 202165 895100 628855 322903 180498 928129 > 2781 [i]