Best Known (137−39, 137, s)-Nets in Base 9
(137−39, 137, 780)-Net over F9 — Constructive and digital
Digital (98, 137, 780)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (8, 27, 40)-net over F9, using
- net from sequence [i] based on digital (8, 39)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 8 and N(F) ≥ 40, using
- net from sequence [i] based on digital (8, 39)-sequence over F9, using
- digital (71, 110, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 55, 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, 55, 370)-net over F81, using
- digital (8, 27, 40)-net over F9, using
(137−39, 137, 5871)-Net over F9 — Digital
Digital (98, 137, 5871)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9137, 5871, F9, 39) (dual of [5871, 5734, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(9137, 6561, F9, 39) (dual of [6561, 6424, 40]-code), using
- an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- discarding factors / shortening the dual code based on linear OA(9137, 6561, F9, 39) (dual of [6561, 6424, 40]-code), using
(137−39, 137, 6706442)-Net in Base 9 — Upper bound on s
There is no (98, 137, 6706443)-net in base 9, because
- 1 times m-reduction [i] would yield (98, 136, 6706443)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 5983 863037 325554 599904 333966 108410 892714 294405 971462 845383 139562 686832 390304 451349 737158 034189 847841 471595 350960 697252 338429 984777 > 9136 [i]