Best Known (40, 64, s)-Nets in Base 9
(40, 64, 344)-Net over F9 — Constructive and digital
Digital (40, 64, 344)-net over F9, using
- 2 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, 64, 600)-Net over F9 — Digital
Digital (40, 64, 600)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(964, 600, F9, 24) (dual of [600, 536, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(964, 728, F9, 24) (dual of [728, 664, 25]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,23], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(964, 728, F9, 24) (dual of [728, 664, 25]-code), using
(40, 64, 81194)-Net in Base 9 — Upper bound on s
There is no (40, 64, 81195)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 11 790808 341033 704632 498894 759841 414175 740757 072433 480372 300065 > 964 [i]