Best Known (88−33, 88, s)-Nets in Base 9
(88−33, 88, 344)-Net over F9 — Constructive and digital
Digital (55, 88, 344)-net over F9, using
- 8 times m-reduction [i] based on digital (55, 96, 344)-net over F9, using
- trace code for nets [i] based on digital (7, 48, 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, 48, 172)-net over F81, using
(88−33, 88, 723)-Net over F9 — Digital
Digital (55, 88, 723)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(988, 723, F9, 33) (dual of [723, 635, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(988, 728, F9, 33) (dual of [728, 640, 34]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,32], and designed minimum distance d ≥ |I|+1 = 34 [i]
- discarding factors / shortening the dual code based on linear OA(988, 728, F9, 33) (dual of [728, 640, 34]-code), using
(88−33, 88, 131253)-Net in Base 9 — Upper bound on s
There is no (55, 88, 131254)-net in base 9, because
- 1 times m-reduction [i] would yield (55, 87, 131254)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 104501 823358 538083 622767 022045 699215 398092 927019 099595 377836 913151 827535 582308 229377 > 987 [i]