Best Known (138−133, 138, s)-Nets in Base 9
(138−133, 138, 32)-Net over F9 — Constructive and digital
Digital (5, 138, 32)-net over F9, using
- net from sequence [i] based on digital (5, 31)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 5 and N(F) ≥ 32, using
(138−133, 138, 54)-Net over F9 — Upper bound on s (digital)
There is no digital (5, 138, 55)-net over F9, because
- 88 times m-reduction [i] would yield digital (5, 50, 55)-net over F9, but
- extracting embedded orthogonal array [i] would yield linear OA(950, 55, F9, 45) (dual of [55, 5, 46]-code), but
- construction Y1 [i] would yield
- OA(949, 51, S9, 45), but
- the (dual) Plotkin bound shows that M ≥ 1 546132 562196 033993 109383 389296 863818 106322 566003 / 23 > 949 [i]
- OA(95, 55, S9, 4), but
- discarding factors would yield OA(95, 44, S9, 4), but
- the Rao or (dual) Hamming bound shows that M ≥ 60897 > 95 [i]
- discarding factors would yield OA(95, 44, S9, 4), but
- OA(949, 51, S9, 45), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(950, 55, F9, 45) (dual of [55, 5, 46]-code), but
(138−133, 138, 56)-Net in Base 9 — Upper bound on s
There is no (5, 138, 57)-net in base 9, because
- 88 times m-reduction [i] would yield (5, 50, 57)-net in base 9, but
- extracting embedded orthogonal array [i] would yield OA(950, 57, S9, 45), but
- the linear programming bound shows that M ≥ 298105 180919 330726 109440 883140 160422 177625 734983 340421 / 537625 > 950 [i]
- extracting embedded orthogonal array [i] would yield OA(950, 57, S9, 45), but