Best Known (137−135, 137, s)-Nets in Base 9
(137−135, 137, 20)-Net over F9 — Constructive and digital
Digital (2, 137, 20)-net over F9, using
- net from sequence [i] based on digital (2, 19)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 2 and N(F) ≥ 20, using
(137−135, 137, 29)-Net over F9 — Upper bound on s (digital)
There is no digital (2, 137, 30)-net over F9, because
- 117 times m-reduction [i] would yield digital (2, 20, 30)-net over F9, but
- extracting embedded orthogonal array [i] would yield linear OA(920, 30, F9, 18) (dual of [30, 10, 19]-code), but
- residual code [i] would yield OA(92, 11, S9, 2), but
- bound for OAs with strength k = 2 [i]
- the Rao or (dual) Hamming bound shows that M ≥ 89 > 92 [i]
- residual code [i] would yield OA(92, 11, S9, 2), but
- extracting embedded orthogonal array [i] would yield linear OA(920, 30, F9, 18) (dual of [30, 10, 19]-code), but
(137−135, 137, 30)-Net in Base 9 — Upper bound on s
There is no (2, 137, 31)-net in base 9, because
- 110 times m-reduction [i] would yield (2, 27, 31)-net in base 9, but
- extracting embedded orthogonal array [i] would yield OA(927, 31, S9, 25), but
- the linear programming bound shows that M ≥ 25644 034018 340666 323462 064529 / 377 > 927 [i]
- extracting embedded orthogonal array [i] would yield OA(927, 31, S9, 25), but