Best Known (7, 7+9, s)-Nets in Base 9
(7, 7+9, 38)-Net over F9 — Constructive and digital
Digital (7, 16, 38)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 10)-net over F9, using
- net from sequence [i] based on digital (0, 9)-sequence over F9, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 0 and N(F) ≥ 10, using
- the rational function field F9(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 9)-sequence over F9, using
- digital (3, 12, 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 (0, 4, 10)-net over F9, using
(7, 7+9, 44)-Net over F9 — Digital
Digital (7, 16, 44)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(916, 44, F9, 9) (dual of [44, 28, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(916, 80, F9, 9) (dual of [80, 64, 10]-code), using
- 2 times truncation [i] based on linear OA(918, 82, F9, 11) (dual of [82, 64, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(918, 81, F9, 11) (dual of [81, 63, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 80 = 92−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(917, 81, F9, 10) (dual of [81, 64, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 80 = 92−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- 2 times truncation [i] based on linear OA(918, 82, F9, 11) (dual of [82, 64, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(916, 80, F9, 9) (dual of [80, 64, 10]-code), using
(7, 7+9, 1046)-Net in Base 9 — Upper bound on s
There is no (7, 16, 1047)-net in base 9, because
- 1 times m-reduction [i] would yield (7, 15, 1047)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 206 656153 411809 > 915 [i]