Best Known (131−37, 131, s)-Nets in Base 9
(131−37, 131, 776)-Net over F9 — Constructive and digital
Digital (94, 131, 776)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (7, 25, 36)-net over F9, using
- net from sequence [i] based on digital (7, 35)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 7 and N(F) ≥ 36, using
- net from sequence [i] based on digital (7, 35)-sequence over F9, using
- 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
- digital (7, 25, 36)-net over F9, using
(131−37, 131, 6068)-Net over F9 — Digital
Digital (94, 131, 6068)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9131, 6068, F9, 37) (dual of [6068, 5937, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(9131, 6571, F9, 37) (dual of [6571, 6440, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(33) [i] based on
- linear OA(9129, 6561, F9, 37) (dual of [6561, 6432, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(9121, 6561, F9, 34) (dual of [6561, 6440, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(92, 10, F9, 2) (dual of [10, 8, 3]-code or 10-arc in PG(1,9)), using
- extended Reed–Solomon code RSe(8,9) [i]
- Hamming code H(2,9) [i]
- construction X applied to Ce(36) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(9131, 6571, F9, 37) (dual of [6571, 6440, 38]-code), using
(131−37, 131, 7358527)-Net in Base 9 — Upper bound on s
There is no (94, 131, 7358528)-net in base 9, because
- 1 times m-reduction [i] would yield (94, 130, 7358528)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 11259 710057 124264 251302 558519 175068 902997 730880 198616 842215 279484 930536 638389 934613 690941 033166 660422 391002 256092 346512 600065 > 9130 [i]