Best Known (54−19, 54, s)-Nets in Base 9
(54−19, 54, 344)-Net over F9 — Constructive and digital
Digital (35, 54, 344)-net over F9, using
- 2 times m-reduction [i] based on digital (35, 56, 344)-net over F9, using
- trace code for nets [i] based on digital (7, 28, 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, 28, 172)-net over F81, using
(54−19, 54, 767)-Net over F9 — Digital
Digital (35, 54, 767)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(954, 767, F9, 19) (dual of [767, 713, 20]-code), using
- 32 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 7 times 0, 1, 20 times 0) [i] based on linear OA(949, 730, F9, 19) (dual of [730, 681, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 730 | 96−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- 32 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 7 times 0, 1, 20 times 0) [i] based on linear OA(949, 730, F9, 19) (dual of [730, 681, 20]-code), using
(54−19, 54, 215813)-Net in Base 9 — Upper bound on s
There is no (35, 54, 215814)-net in base 9, because
- 1 times m-reduction [i] would yield (35, 53, 215814)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 375 712977 490327 723497 898642 746672 268675 548486 194865 > 953 [i]