Best Known (81, 81+30, s)-Nets in Base 9
(81, 81+30, 770)-Net over F9 — Constructive and digital
Digital (81, 111, 770)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (4, 19, 30)-net over F9, using
- net from sequence [i] based on digital (4, 29)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 4 and N(F) ≥ 30, using
- net from sequence [i] based on digital (4, 29)-sequence over F9, using
- digital (62, 92, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 46, 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
- trace code for nets [i] based on digital (16, 46, 370)-net over F81, using
- digital (4, 19, 30)-net over F9, using
(81, 81+30, 6583)-Net over F9 — Digital
Digital (81, 111, 6583)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9111, 6583, F9, 30) (dual of [6583, 6472, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(24) [i] based on
- linear OA(9105, 6561, F9, 30) (dual of [6561, 6456, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(989, 6561, F9, 25) (dual of [6561, 6472, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(96, 22, F9, 4) (dual of [22, 16, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(96, 72, F9, 4) (dual of [72, 66, 5]-code), using
- 1 times truncation [i] based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(96, 72, F9, 4) (dual of [72, 66, 5]-code), using
- construction X applied to Ce(29) ⊂ Ce(24) [i] based on
(81, 81+30, large)-Net in Base 9 — Upper bound on s
There is no (81, 111, large)-net in base 9, because
- 28 times m-reduction [i] would yield (81, 83, large)-net in base 9, but