Best Known (34−30, 34, s)-Nets in Base 9
(34−30, 34, 30)-Net over F9 — Constructive and digital
Digital (4, 34, 30)-net over F9, using
- net from sequence [i] based on digital (4, 29)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 4 and N(F) ≥ 30, using
(34−30, 34, 84)-Net over F9 — Upper bound on s (digital)
There is no digital (4, 34, 85)-net over F9, because
- 3 times m-reduction [i] would yield digital (4, 31, 85)-net over F9, but
- extracting embedded orthogonal array [i] would yield linear OA(931, 85, F9, 27) (dual of [85, 54, 28]-code), but
- construction Y1 [i] would yield
- linear OA(930, 37, F9, 27) (dual of [37, 7, 28]-code), but
- construction Y1 [i] would yield
- OA(929, 31, S9, 27), but
- the (dual) Plotkin bound shows that M ≥ 42391 158275 216203 514294 433201 / 7 > 929 [i]
- OA(97, 37, S9, 6), but
- the linear programming bound shows that M ≥ 8306 954271 / 1613 > 97 [i]
- OA(929, 31, S9, 27), but
- construction Y1 [i] would yield
- OA(954, 85, S9, 48), but
- discarding factors would yield OA(954, 83, S9, 48), but
- the linear programming bound shows that M ≥ 10132 927410 689078 360872 417149 617665 028299 664062 593121 142935 280521 657389 / 2 927120 765949 062837 > 954 [i]
- discarding factors would yield OA(954, 83, S9, 48), but
- linear OA(930, 37, F9, 27) (dual of [37, 7, 28]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(931, 85, F9, 27) (dual of [85, 54, 28]-code), but
(34−30, 34, 87)-Net in Base 9 — Upper bound on s
There is no (4, 34, 88)-net in base 9, because
- extracting embedded orthogonal array [i] would yield OA(934, 88, S9, 30), but
- the linear programming bound shows that M ≥ 5 291150 204580 024893 248614 264982 244147 945665 636668 / 18840 733210 265293 > 934 [i]