Best Known (56−19, 56, s)-Nets in Base 9
(56−19, 56, 344)-Net over F9 — Constructive and digital
Digital (37, 56, 344)-net over F9, using
- 4 times m-reduction [i] based on digital (37, 60, 344)-net over F9, using
- trace code for nets [i] based on digital (7, 30, 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, 30, 172)-net over F81, using
(56−19, 56, 897)-Net over F9 — Digital
Digital (37, 56, 897)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(956, 897, F9, 19) (dual of [897, 841, 20]-code), using
- 160 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 7 times 0, 1, 20 times 0, 1, 47 times 0, 1, 79 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]
- 160 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 7 times 0, 1, 20 times 0, 1, 47 times 0, 1, 79 times 0) [i] based on linear OA(949, 730, F9, 19) (dual of [730, 681, 20]-code), using
(56−19, 56, 351671)-Net in Base 9 — Upper bound on s
There is no (37, 56, 351672)-net in base 9, because
- 1 times m-reduction [i] would yield (37, 55, 351672)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 30432 941624 260114 613637 531282 758062 735419 987546 191809 > 955 [i]