Best Known (20, 20+20, s)-Nets in Base 27
(20, 20+20, 140)-Net over F27 — Constructive and digital
Digital (20, 40, 140)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (4, 14, 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, 26, 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, 14, 64)-net over F27, using
(20, 20+20, 172)-Net in Base 27 — Constructive
(20, 40, 172)-net in base 27, using
- 12 times m-reduction [i] based on (20, 52, 172)-net in base 27, using
- base change [i] based on digital (7, 39, 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, 39, 172)-net over F81, using
(20, 20+20, 367)-Net over F27 — Digital
Digital (20, 40, 367)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2740, 367, F27, 2, 20) (dual of [(367, 2), 694, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2740, 734, F27, 20) (dual of [734, 694, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- linear OA(2739, 729, F27, 20) (dual of [729, 690, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- 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(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(19) ⊂ Ce(17) [i] based on
- OOA 2-folding [i] based on linear OA(2740, 734, F27, 20) (dual of [734, 694, 21]-code), using
(20, 20+20, 92562)-Net in Base 27 — Upper bound on s
There is no (20, 40, 92563)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 1797 106587 670773 255049 011293 448822 634614 643391 634610 171653 > 2740 [i]