Best Known (10−3, 10, s)-Nets in Base 7
(10−3, 10, 122500)-Net over F7 — Constructive and digital
Digital (7, 10, 122500)-net over F7, using
- net defined by OOA [i] based on linear OOA(710, 122500, F7, 3, 3) (dual of [(122500, 3), 367490, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(710, 122500, F7, 2, 3) (dual of [(122500, 2), 244990, 4]-NRT-code), using
(10−3, 10, 3158440)-Net over F7 — Upper bound on s (digital)
There is no digital (7, 10, 3158441)-net over F7, because
- extracting embedded orthogonal array [i] would yield linear OA(710, 3158441, F7, 3) (dual of [3158441, 3158431, 4]-code or 3158441-cap in PG(9,7)), 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)
- 409252-cap in AG(8,7), but
- 4 times the recursive bound from Bierbrauer and Edel [i] would yield 239-cap in AG(4,7) (see above)
- 2674866-cap in AG(9,7), but
- 5 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
(10−3, 10, 6725600)-Net in Base 7 — Upper bound on s
There is no (7, 10, 6725601)-net in base 7, because
- extracting embedded orthogonal array [i] would yield OA(710, 6725601, S7, 3), but