Best Known (72, 99, s)-Nets in Base 9
(72, 99, 750)-Net over F9 — Constructive and digital
Digital (72, 99, 750)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (0, 13, 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 (59, 86, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 43, 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, 43, 370)-net over F81, using
- digital (0, 13, 10)-net over F9, using
(72, 99, 6573)-Net over F9 — Digital
Digital (72, 99, 6573)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(999, 6573, F9, 27) (dual of [6573, 6474, 28]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(998, 6571, F9, 27) (dual of [6571, 6473, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(997, 6562, F9, 27) (dual of [6562, 6465, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 98−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(989, 6562, F9, 25) (dual of [6562, 6473, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 98−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(91, 9, F9, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(998, 6572, F9, 26) (dual of [6572, 6474, 27]-code), using Gilbert–Varšamov bound and bm = 998 > Vbs−1(k−1) = 642 311279 648027 383432 106219 116785 536272 196654 627827 804563 939633 261225 456648 603622 208106 440985 [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
- linear OA(998, 6571, F9, 27) (dual of [6571, 6473, 28]-code), using
- construction X with Varšamov bound [i] based on
(72, 99, large)-Net in Base 9 — Upper bound on s
There is no (72, 99, large)-net in base 9, because
- 25 times m-reduction [i] would yield (72, 74, large)-net in base 9, but