Best Known (18, 18+18, s)-Nets in Base 27
(18, 18+18, 132)-Net over F27 — Constructive and digital
Digital (18, 36, 132)-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 (5, 23, 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, 13, 64)-net over F27, using
(18, 18+18, 172)-Net in Base 27 — Constructive
(18, 36, 172)-net in base 27, using
- 8 times m-reduction [i] based on (18, 44, 172)-net in base 27, using
- base change [i] based on digital (7, 33, 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, 33, 172)-net over F81, using
(18, 18+18, 367)-Net over F27 — Digital
Digital (18, 36, 367)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2736, 367, F27, 2, 18) (dual of [(367, 2), 698, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2736, 734, F27, 18) (dual of [734, 698, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [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(2731, 729, F27, 16) (dual of [729, 698, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(271, 5, F27, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- OOA 2-folding [i] based on linear OA(2736, 734, F27, 18) (dual of [734, 698, 19]-code), using
(18, 18+18, 84763)-Net in Base 27 — Upper bound on s
There is no (18, 36, 84764)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 3381 430666 004458 789436 202864 848745 059303 993191 475033 > 2736 [i]