Best Known (5, 8, s)-Nets in Base 7
(5, 8, 6472)-Net over F7 — Constructive and digital
Digital (5, 8, 6472)-net over F7, using
- net defined by OOA [i] based on linear OOA(78, 6472, F7, 3, 3) (dual of [(6472, 3), 19408, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(78, 6472, F7, 2, 3) (dual of [(6472, 2), 12936, 4]-NRT-code), using
(5, 8, 74324)-Net over F7 — Upper bound on s (digital)
There is no digital (5, 8, 74325)-net over F7, because
- extracting embedded orthogonal array [i] would yield linear OA(78, 74325, F7, 3) (dual of [74325, 74317, 4]-code or 74325-cap in PG(7,7)), but
- removing affine subspaces [i] would yield
- linear OA(76, 1659, F7, 3) (dual of [1659, 1653, 4]-code or 1659-cap in PG(5,7)), but
- 9735-cap in AG(6,7), but
- 2 times the recursive bound from Bierbrauer and Edel [i] would yield 239-cap in AG(4,7), but
- 62933-cap in AG(7,7), but
- 3 times the recursive bound from Bierbrauer and Edel [i] would yield 239-cap in AG(4,7) (see above)
- removing affine subspaces [i] would yield
(5, 8, 137256)-Net in Base 7 — Upper bound on s
There is no (5, 8, 137257)-net in base 7, because
- extracting embedded orthogonal array [i] would yield OA(78, 137257, S7, 3), but