Best Known (82−31, 82, s)-Nets in Base 9
(82−31, 82, 344)-Net over F9 — Constructive and digital
Digital (51, 82, 344)-net over F9, using
- 6 times m-reduction [i] based on digital (51, 88, 344)-net over F9, using
- trace code for nets [i] based on digital (7, 44, 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, 44, 172)-net over F81, using
(82−31, 82, 659)-Net over F9 — Digital
Digital (51, 82, 659)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(982, 659, F9, 31) (dual of [659, 577, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(982, 728, F9, 31) (dual of [728, 646, 32]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,30], and designed minimum distance d ≥ |I|+1 = 32 [i]
- discarding factors / shortening the dual code based on linear OA(982, 728, F9, 31) (dual of [728, 646, 32]-code), using
(82−31, 82, 114170)-Net in Base 9 — Upper bound on s
There is no (51, 82, 114171)-net in base 9, because
- 1 times m-reduction [i] would yield (51, 81, 114171)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 196649 854778 681391 409846 717737 602881 634658 541788 933360 996486 709592 109726 423081 > 981 [i]