Best Known (33, 33+18, s)-Nets in Base 9
(33, 33+18, 344)-Net over F9 — Constructive and digital
Digital (33, 51, 344)-net over F9, using
- 1 times m-reduction [i] based on digital (33, 52, 344)-net over F9, using
- trace code for nets [i] based on digital (7, 26, 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, 26, 172)-net over F81, using
(33, 33+18, 752)-Net over F9 — Digital
Digital (33, 51, 752)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(951, 752, F9, 18) (dual of [752, 701, 19]-code), using
- 20 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 14 times 0) [i] based on linear OA(948, 729, F9, 18) (dual of [729, 681, 19]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(949, 730, F9, 19) (dual of [730, 681, 20]-code), using
- 20 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 14 times 0) [i] based on linear OA(948, 729, F9, 18) (dual of [729, 681, 19]-code), using
(33, 33+18, 132439)-Net in Base 9 — Upper bound on s
There is no (33, 51, 132440)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 4 638426 146537 560880 812026 283706 766494 856320 003265 > 951 [i]