Best Known (61, 61+37, s)-Nets in Base 9
(61, 61+37, 344)-Net over F9 — Constructive and digital
Digital (61, 98, 344)-net over F9, using
- 10 times m-reduction [i] based on digital (61, 108, 344)-net over F9, using
- trace code for nets [i] based on digital (7, 54, 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, 54, 172)-net over F81, using
(61, 61+37, 741)-Net over F9 — Digital
Digital (61, 98, 741)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(998, 741, F9, 37) (dual of [741, 643, 38]-code), using
- 10 step Varšamov–Edel lengthening with (ri) = (1, 9 times 0) [i] based on linear OA(997, 730, F9, 37) (dual of [730, 633, 38]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 730 | 96−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- 10 step Varšamov–Edel lengthening with (ri) = (1, 9 times 0) [i] based on linear OA(997, 730, F9, 37) (dual of [730, 633, 38]-code), using
(61, 61+37, 131012)-Net in Base 9 — Upper bound on s
There is no (61, 98, 131013)-net in base 9, because
- 1 times m-reduction [i] would yield (61, 97, 131013)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 364 400682 514013 576945 842998 149400 326839 945766 268506 842937 601892 633838 562741 211246 950360 949905 > 997 [i]