Best Known (37−33, 37, s)-Nets in Base 9
(37−33, 37, 30)-Net over F9 — Constructive and digital
Digital (4, 37, 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
(37−33, 37, 49)-Net over F9 — Upper bound on s (digital)
There is no digital (4, 37, 50)-net over F9, because
- 1 times m-reduction [i] would yield digital (4, 36, 50)-net over F9, but
- extracting embedded orthogonal array [i] would yield linear OA(936, 50, F9, 32) (dual of [50, 14, 33]-code), but
- construction Y1 [i] would yield
- linear OA(935, 38, F9, 32) (dual of [38, 3, 33]-code), but
- “Mas†bound on codes from Brouwer’s database [i]
- linear OA(914, 50, F9, 12) (dual of [50, 36, 13]-code), but
- discarding factors / shortening the dual code would yield linear OA(914, 44, F9, 12) (dual of [44, 30, 13]-code), but
- construction Y1 [i] would yield
- linear OA(913, 17, F9, 12) (dual of [17, 4, 13]-code), but
- “MPa†bound on codes from Brouwer’s database [i]
- linear OA(930, 44, F9, 27) (dual of [44, 14, 28]-code), but
- discarding factors / shortening the dual code would yield linear OA(930, 39, F9, 27) (dual of [39, 9, 28]-code), but
- residual code [i] would yield OA(93, 11, S9, 3), but
- discarding factors / shortening the dual code would yield linear OA(930, 39, F9, 27) (dual of [39, 9, 28]-code), but
- linear OA(913, 17, F9, 12) (dual of [17, 4, 13]-code), but
- construction Y1 [i] would yield
- discarding factors / shortening the dual code would yield linear OA(914, 44, F9, 12) (dual of [44, 30, 13]-code), but
- linear OA(935, 38, F9, 32) (dual of [38, 3, 33]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(936, 50, F9, 32) (dual of [50, 14, 33]-code), but
(37−33, 37, 77)-Net in Base 9 — Upper bound on s
There is no (4, 37, 78)-net in base 9, because
- extracting embedded orthogonal array [i] would yield OA(937, 78, S9, 33), but
- the linear programming bound shows that M ≥ 199356 054373 293137 005126 556242 150856 557793 215196 056979 783877 / 900916 662343 931597 492741 > 937 [i]