Best Known (40, 63, s)-Nets in Base 9
(40, 63, 344)-Net over F9 — Constructive and digital
Digital (40, 63, 344)-net over F9, using
- 3 times m-reduction [i] based on digital (40, 66, 344)-net over F9, using
- trace code for nets [i] based on digital (7, 33, 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
- trace code for nets [i] based on digital (7, 33, 172)-net over F81, using
(40, 63, 701)-Net over F9 — Digital
Digital (40, 63, 701)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(963, 701, F9, 23) (dual of [701, 638, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(963, 728, F9, 23) (dual of [728, 665, 24]-code), using
- the primitive narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- discarding factors / shortening the dual code based on linear OA(963, 728, F9, 23) (dual of [728, 665, 24]-code), using
(40, 63, 146677)-Net in Base 9 — Upper bound on s
There is no (40, 63, 146678)-net in base 9, because
- 1 times m-reduction [i] would yield (40, 62, 146678)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 145564 856982 664860 676039 080849 102166 322948 489763 270013 603665 > 962 [i]