Best Known (80−22, 80, s)-Nets in Base 9
(80−22, 80, 740)-Net over F9 — Constructive and digital
Digital (58, 80, 740)-net over F9, using
- 4 times m-reduction [i] based on digital (58, 84, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 42, 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, 42, 370)-net over F81, using
(80−22, 80, 6091)-Net over F9 — Digital
Digital (58, 80, 6091)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(980, 6091, F9, 22) (dual of [6091, 6011, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(980, 6576, F9, 22) (dual of [6576, 6496, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- linear OA(977, 6561, F9, 22) (dual of [6561, 6484, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(965, 6561, F9, 19) (dual of [6561, 6496, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(93, 15, F9, 2) (dual of [15, 12, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 92−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(980, 6576, F9, 22) (dual of [6576, 6496, 23]-code), using
(80−22, 80, 5344033)-Net in Base 9 — Upper bound on s
There is no (58, 80, 5344034)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 21847 471218 566643 237143 700286 657753 506507 141062 920245 195635 159775 773027 201265 > 980 [i]