Best Known (31−28, 31, s)-Nets in Base 9
(31−28, 31, 28)-Net over F9 — Constructive and digital
Digital (3, 31, 28)-net over F9, using
- net from sequence [i] based on digital (3, 27)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 3 and N(F) ≥ 28, using
- the Hermitian function field over F9 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 3 and N(F) ≥ 28, using
(31−28, 31, 36)-Net over F9 — Upper bound on s (digital)
There is no digital (3, 31, 37)-net over F9, because
- 1 times m-reduction [i] would yield digital (3, 30, 37)-net over F9, but
- extracting embedded orthogonal array [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
- extracting embedded orthogonal array [i] would yield linear OA(930, 37, F9, 27) (dual of [37, 7, 28]-code), but
(31−28, 31, 62)-Net in Base 9 — Upper bound on s
There is no (3, 31, 63)-net in base 9, because
- extracting embedded orthogonal array [i] would yield OA(931, 63, S9, 28), but
- the linear programming bound shows that M ≥ 6759 171149 749488 852719 246829 729701 066529 / 16771 744361 > 931 [i]