Best Known (78, 78+29, s)-Nets in Base 9
(78, 78+29, 768)-Net over F9 — Constructive and digital
Digital (78, 107, 768)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (3, 17, 28)-net over F9, using
- net from sequence [i] based on digital (3, 27)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 3 and N(F) ≥ 28, using
- the Hermitian function field over F9 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 3 and N(F) ≥ 28, using
- net from sequence [i] based on digital (3, 27)-sequence over F9, using
- digital (61, 90, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 45, 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, 45, 370)-net over F81, using
- digital (3, 17, 28)-net over F9, using
(78, 78+29, 6583)-Net over F9 — Digital
Digital (78, 107, 6583)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9107, 6583, F9, 29) (dual of [6583, 6476, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(23) [i] based on
- linear OA(9101, 6561, F9, 29) (dual of [6561, 6460, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(985, 6561, F9, 24) (dual of [6561, 6476, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(96, 22, F9, 4) (dual of [22, 16, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(96, 72, F9, 4) (dual of [72, 66, 5]-code), using
- 1 times truncation [i] based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(96, 72, F9, 4) (dual of [72, 66, 5]-code), using
- construction X applied to Ce(28) ⊂ Ce(23) [i] based on
(78, 78+29, large)-Net in Base 9 — Upper bound on s
There is no (78, 107, large)-net in base 9, because
- 27 times m-reduction [i] would yield (78, 80, large)-net in base 9, but