Best Known (89−25, 89, s)-Nets in Base 9
(89−25, 89, 740)-Net over F9 — Constructive and digital
Digital (64, 89, 740)-net over F9, using
- 7 times m-reduction [i] based on digital (64, 96, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 48, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 48, 370)-net over F81, using
(89−25, 89, 5264)-Net over F9 — Digital
Digital (64, 89, 5264)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(989, 5264, F9, 25) (dual of [5264, 5175, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(989, 6561, F9, 25) (dual of [6561, 6472, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(989, 6561, F9, 25) (dual of [6561, 6472, 26]-code), using
(89−25, 89, 6577325)-Net in Base 9 — Upper bound on s
There is no (64, 89, 6577326)-net in base 9, because
- 1 times m-reduction [i] would yield (64, 88, 6577326)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 940461 108397 738608 900648 936910 484040 341960 018788 945971 277579 223095 609926 150399 348161 > 988 [i]