Best Known (81−23, 81, s)-Nets in Base 9
(81−23, 81, 740)-Net over F9 — Constructive and digital
Digital (58, 81, 740)-net over F9, using
- 3 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
(81−23, 81, 4672)-Net over F9 — Digital
Digital (58, 81, 4672)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(981, 4672, F9, 23) (dual of [4672, 4591, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(981, 6561, F9, 23) (dual of [6561, 6480, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(981, 6561, F9, 23) (dual of [6561, 6480, 24]-code), using
(81−23, 81, 5344033)-Net in Base 9 — Upper bound on s
There is no (58, 81, 5344034)-net in base 9, because
- 1 times m-reduction [i] would yield (58, 80, 5344034)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 21847 471218 566643 237143 700286 657753 506507 141062 920245 195635 159775 773027 201265 > 980 [i]