Best Known (31−27, 31, s)-Nets in Base 9
(31−27, 31, 30)-Net over F9 — Constructive and digital
Digital (4, 31, 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
(31−27, 31, 84)-Net over F9 — Upper bound on s (digital)
There is no digital (4, 31, 85)-net over F9, because
- 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
(31−27, 31, 96)-Net in Base 9 — Upper bound on s
There is no (4, 31, 97)-net in base 9, because
- extracting embedded orthogonal array [i] would yield OA(931, 97, S9, 27), but
- the linear programming bound shows that M ≥ 6 768259 797617 940013 515376 710477 747084 076431 301875 / 17 682194 234509 296851 > 931 [i]