Best Known (69, 69+28, s)-Nets in Base 9
(69, 69+28, 740)-Net over F9 — Constructive and digital
Digital (69, 97, 740)-net over F9, using
- 9 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+28, 4386)-Net over F9 — Digital
Digital (69, 97, 4386)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(997, 4386, F9, 28) (dual of [4386, 4289, 29]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(997, 6561, F9, 28) (dual of [6561, 6464, 29]-code), using
(69, 69+28, 3089612)-Net in Base 9 — Upper bound on s
There is no (69, 97, 3089613)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 364 355117 990510 734224 999268 735341 889905 106423 433319 837558 795575 958709 417576 945834 132009 955313 > 997 [i]