Best Known (5, 8, s)-Nets in Base 4
(5, 8, 756)-Net over F4 — Constructive and digital
Digital (5, 8, 756)-net over F4, using
- net defined by OOA [i] based on linear OOA(48, 756, F4, 3, 3) (dual of [(756, 3), 2260, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(48, 756, F4, 2, 3) (dual of [(756, 2), 1504, 4]-NRT-code), using
(5, 8, 2358)-Net over F4 — Upper bound on s (digital)
There is no digital (5, 8, 2359)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(48, 2359, F4, 3) (dual of [2359, 2351, 4]-code or 2359-cap in PG(7,4)), 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
- 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
(5, 8, 5460)-Net in Base 4 — Upper bound on s
There is no (5, 8, 5461)-net in base 4, because
- extracting embedded orthogonal array [i] would yield OA(48, 5461, S4, 3), but