Best Known (4, 7, s)-Nets in Base 8
(4, 7, 4224)-Net over F8 — Constructive and digital
Digital (4, 7, 4224)-net over F8, using
- net defined by OOA [i] based on linear OOA(87, 4224, F8, 3, 3) (dual of [(4224, 3), 12665, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(87, 4224, F8, 2, 3) (dual of [(4224, 2), 8441, 4]-NRT-code), using
(4, 7, 27649)-Net over F8 — Upper bound on s (digital)
There is no digital (4, 7, 27650)-net over F8, because
- extracting embedded orthogonal array [i] would yield linear OA(87, 27650, F8, 3) (dual of [27650, 27643, 4]-code or 27650-cap in PG(6,8)), but
- removing affine subspaces [i] would yield
- linear OA(85, 494, F8, 3) (dual of [494, 489, 4]-code or 494-cap in PG(4,8)), but
- 3284-cap in AG(5,8), but
- 2 times the recursive bound from Bierbrauer and Edel [i] would yield 65-cap in AG(3,8), but
- 23874-cap in AG(6,8), but
- 3 times the recursive bound from Bierbrauer and Edel [i] would yield 65-cap in AG(3,8) (see above)
- removing affine subspaces [i] would yield
(4, 7, 37448)-Net in Base 8 — Upper bound on s
There is no (4, 7, 37449)-net in base 8, because
- extracting embedded orthogonal array [i] would yield OA(87, 37449, S8, 3), but