Best Known (7, 7+3, s)-Nets in Base 4
(7, 7+3, 5040)-Net over F4 — Constructive and digital
Digital (7, 10, 5040)-net over F4, using
- net defined by OOA [i] based on linear OOA(410, 5040, F4, 3, 3) (dual of [(5040, 3), 15110, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(410, 5040, F4, 2, 3) (dual of [(5040, 2), 10070, 4]-NRT-code), using
(7, 7+3, 31770)-Net over F4 — Upper bound on s (digital)
There is no digital (7, 10, 31771)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(410, 31771, F4, 3) (dual of [31771, 31761, 4]-code or 31771-cap in PG(9,4)), but
- removing affine subspaces [i] would yield
- linear OA(47, 608, F4, 3) (dual of [608, 601, 4]-code or 608-cap in PG(6,4)), but
- 1 times Hill recurrence [i] would yield linear OA(46, 154, F4, 3) (dual of [154, 148, 4]-code or 154-cap in PG(5,4)), but
- construction Y1 [i] would yield
- linear OA(45, 42, F4, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,4)), but
- linear OA(4148, 154, F4, 112) (dual of [154, 6, 113]-code), but
- discarding factors / shortening the dual code would yield linear OA(4148, 153, F4, 112) (dual of [153, 5, 113]-code), but
- residual code [i] would yield linear OA(436, 40, F4, 28) (dual of [40, 4, 29]-code), but
- residual code [i] would yield linear OA(48, 11, F4, 7) (dual of [11, 3, 8]-code), but
- residual code [i] would yield linear OA(436, 40, F4, 28) (dual of [40, 4, 29]-code), but
- discarding factors / shortening the dual code would yield linear OA(4148, 153, F4, 112) (dual of [153, 5, 113]-code), but
- construction Y1 [i] would yield
- 1 times Hill recurrence [i] would yield linear OA(46, 154, F4, 3) (dual of [154, 148, 4]-code or 154-cap in PG(5,4)), but
- 1752-cap in AG(7,4), but
- 3 times the recursive bound from Bierbrauer and Edel [i] would yield 41-cap in AG(4,4), but
- 6329-cap in AG(8,4), but
- 4 times the recursive bound from Bierbrauer and Edel [i] would yield 41-cap in AG(4,4) (see above)
- 23085-cap in AG(9,4), but
- 5 times the recursive bound from Bierbrauer and Edel [i] would yield 41-cap in AG(4,4) (see above)
- linear OA(47, 608, F4, 3) (dual of [608, 601, 4]-code or 608-cap in PG(6,4)), but
- removing affine subspaces [i] would yield
(7, 7+3, 87380)-Net in Base 4 — Upper bound on s
There is no (7, 10, 87381)-net in base 4, because
- extracting embedded orthogonal array [i] would yield OA(410, 87381, S4, 3), but