Best Known (72−19, 72, s)-Nets in Base 9
(72−19, 72, 740)-Net over F9 — Constructive and digital
Digital (53, 72, 740)-net over F9, using
- 2 times m-reduction [i] based on digital (53, 74, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 37, 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, 37, 370)-net over F81, using
(72−19, 72, 6588)-Net over F9 — Digital
Digital (53, 72, 6588)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(972, 6588, F9, 19) (dual of [6588, 6516, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
- linear OA(965, 6561, F9, 19) (dual of [6561, 6496, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(945, 6561, F9, 13) (dual of [6561, 6516, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(97, 27, F9, 5) (dual of [27, 20, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
(72−19, 72, large)-Net in Base 9 — Upper bound on s
There is no (53, 72, large)-net in base 9, because
- 17 times m-reduction [i] would yield (53, 55, large)-net in base 9, but