Best Known (69−36, 69, s)-Nets in Base 27
(69−36, 69, 166)-Net over F27 — Constructive and digital
Digital (33, 69, 166)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 25, 82)-net over F27, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 7 and N(F) ≥ 82, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- digital (8, 44, 84)-net over F27, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 8 and N(F) ≥ 84, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
- digital (7, 25, 82)-net over F27, using
(69−36, 69, 224)-Net in Base 27 — Constructive
(33, 69, 224)-net in base 27, using
- 11 times m-reduction [i] based on (33, 80, 224)-net in base 27, using
- base change [i] based on digital (13, 60, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- base change [i] based on digital (13, 60, 224)-net over F81, using
(69−36, 69, 367)-Net over F27 — Digital
Digital (33, 69, 367)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2769, 367, F27, 2, 36) (dual of [(367, 2), 665, 37]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2769, 734, F27, 36) (dual of [734, 665, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(33) [i] based on
- linear OA(2768, 729, F27, 36) (dual of [729, 661, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [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(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(35) ⊂ Ce(33) [i] based on
- OOA 2-folding [i] based on linear OA(2769, 734, F27, 36) (dual of [734, 665, 37]-code), using
(69−36, 69, 89126)-Net in Base 27 — Upper bound on s
There is no (33, 69, 89127)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 580 968558 720985 225922 163306 792071 797451 412791 701908 433106 752899 213536 632572 894108 990991 878831 328765 > 2769 [i]