Best Known (99−37, 99, s)-Nets in Base 9
(99−37, 99, 344)-Net over F9 — Constructive and digital
Digital (62, 99, 344)-net over F9, using
- 11 times m-reduction [i] based on digital (62, 110, 344)-net over F9, using
- trace code for nets [i] based on digital (7, 55, 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, 55, 172)-net over F81, using
(99−37, 99, 772)-Net over F9 — Digital
Digital (62, 99, 772)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(999, 772, F9, 37) (dual of [772, 673, 38]-code), using
- 40 step Varšamov–Edel lengthening with (ri) = (1, 9 times 0, 1, 29 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]
- 40 step Varšamov–Edel lengthening with (ri) = (1, 9 times 0, 1, 29 times 0) [i] based on linear OA(997, 730, F9, 37) (dual of [730, 633, 38]-code), using
(99−37, 99, 148022)-Net in Base 9 — Upper bound on s
There is no (62, 99, 148023)-net in base 9, because
- 1 times m-reduction [i] would yield (62, 98, 148023)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 3279 231434 674008 060391 974881 166018 771121 849495 734007 879372 891917 205503 450913 943443 666795 960305 > 998 [i]