Best Known (99, 99+39, s)-Nets in Base 9
(99, 99+39, 780)-Net over F9 — Constructive and digital
Digital (99, 138, 780)-net over F9, using
- 91 times duplication [i] based on digital (98, 137, 780)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (8, 27, 40)-net over F9, using
- net from sequence [i] based on digital (8, 39)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 8 and N(F) ≥ 40, using
- net from sequence [i] based on digital (8, 39)-sequence over F9, using
- digital (71, 110, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 55, 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, 55, 370)-net over F81, using
- digital (8, 27, 40)-net over F9, using
- (u, u+v)-construction [i] based on
(99, 99+39, 6231)-Net over F9 — Digital
Digital (99, 138, 6231)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9138, 6231, F9, 39) (dual of [6231, 6093, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(9138, 6571, F9, 39) (dual of [6571, 6433, 40]-code), using
- construction X applied to C([0,19]) ⊂ C([0,18]) [i] based on
- linear OA(9137, 6562, F9, 39) (dual of [6562, 6425, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 98−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- linear OA(9129, 6562, F9, 37) (dual of [6562, 6433, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 98−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (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([0,19]) ⊂ C([0,18]) [i] based on
- discarding factors / shortening the dual code based on linear OA(9138, 6571, F9, 39) (dual of [6571, 6433, 40]-code), using
(99, 99+39, 7528623)-Net in Base 9 — Upper bound on s
There is no (99, 138, 7528624)-net in base 9, because
- 1 times m-reduction [i] would yield (99, 137, 7528624)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 53854 766564 979416 208604 166642 198961 133040 363841 398524 212685 622470 941717 058639 634161 174109 199440 118696 983773 506441 521902 713398 008449 > 9137 [i]