Best Known (34−13, 34, s)-Nets in Base 9
(34−13, 34, 300)-Net over F9 — Constructive and digital
Digital (21, 34, 300)-net over F9, using
- trace code for nets [i] based on digital (4, 17, 150)-net over F81, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 4 and N(F) ≥ 150, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
(34−13, 34, 442)-Net over F9 — Digital
Digital (21, 34, 442)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(934, 442, F9, 13) (dual of [442, 408, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(934, 728, F9, 13) (dual of [728, 694, 14]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(934, 728, F9, 13) (dual of [728, 694, 14]-code), using
(34−13, 34, 66289)-Net in Base 9 — Upper bound on s
There is no (21, 34, 66290)-net in base 9, because
- 1 times m-reduction [i] would yield (21, 33, 66290)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 30 904611 417895 903727 553906 840225 > 933 [i]