Best Known (5, 5+39, s)-Nets in Base 9
(5, 5+39, 32)-Net over F9 — Constructive and digital
Digital (5, 44, 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
(5, 5+39, 76)-Net over F9 — Upper bound on s (digital)
There is no digital (5, 44, 77)-net over F9, because
- 3 times m-reduction [i] would yield digital (5, 41, 77)-net over F9, but
- extracting embedded orthogonal array [i] would yield linear OA(941, 77, F9, 36) (dual of [77, 36, 37]-code), but
- construction Y1 [i] would yield
- linear OA(940, 45, F9, 36) (dual of [45, 5, 37]-code), but
- construction Y1 [i] would yield
- OA(939, 41, S9, 36), but
- the (dual) Plotkin bound shows that M ≥ 739 044147 071729 616580 416051 031916 488005 / 37 > 939 [i]
- OA(95, 45, 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(939, 41, S9, 36), but
- construction Y1 [i] would yield
- linear OA(936, 77, F9, 32) (dual of [77, 41, 33]-code), but
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code would yield linear OA(936, 50, F9, 32) (dual of [50, 14, 33]-code), but
- linear OA(940, 45, F9, 36) (dual of [45, 5, 37]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(941, 77, F9, 36) (dual of [77, 36, 37]-code), but
(5, 5+39, 90)-Net in Base 9 — Upper bound on s
There is no (5, 44, 91)-net in base 9, because
- extracting embedded orthogonal array [i] would yield OA(944, 91, S9, 39), but
- the linear programming bound shows that M ≥ 2071 706836 797575 615543 016650 958636 720536 640747 872302 474970 623446 256472 073666 / 2134 901384 199222 888941 044905 091615 > 944 [i]