Best Known (54−20, 54, s)-Nets in Base 9
(54−20, 54, 344)-Net over F9 — Constructive and digital
Digital (34, 54, 344)-net over F9, using
- trace code for nets [i] based on digital (7, 27, 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
(54−20, 54, 600)-Net over F9 — Digital
Digital (34, 54, 600)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(954, 600, F9, 20) (dual of [600, 546, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(954, 728, F9, 20) (dual of [728, 674, 21]-code), using
- the primitive narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(954, 728, F9, 20) (dual of [728, 674, 21]-code), using
(54−20, 54, 80494)-Net in Base 9 — Upper bound on s
There is no (34, 54, 80495)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 3381 707928 734121 913152 275340 585187 097845 463509 183665 > 954 [i]