Best Known (69, 69+27, s)-Nets in Base 9
(69, 69+27, 740)-Net over F9 — Constructive and digital
Digital (69, 96, 740)-net over F9, using
- 10 times m-reduction [i] based on digital (69, 106, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 53, 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, 53, 370)-net over F81, using
(69, 69+27, 5364)-Net over F9 — Digital
Digital (69, 96, 5364)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(996, 5364, F9, 27) (dual of [5364, 5268, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(996, 6560, F9, 27) (dual of [6560, 6464, 28]-code), using
- 1 times truncation [i] based on linear OA(997, 6561, F9, 28) (dual of [6561, 6464, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- 1 times truncation [i] based on linear OA(997, 6561, F9, 28) (dual of [6561, 6464, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(996, 6560, F9, 27) (dual of [6560, 6464, 28]-code), using
(69, 69+27, 6662333)-Net in Base 9 — Upper bound on s
There is no (69, 96, 6662334)-net in base 9, because
- 1 times m-reduction [i] would yield (69, 95, 6662334)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 4 498204 459328 716459 767003 342301 508309 470788 817048 284301 027830 819354 431853 203243 064028 213425 > 995 [i]