Best Known (11, 11+9, s)-Nets in Base 2
(11, 11+9, 26)-Net over F2 — Constructive and digital
Digital (11, 20, 26)-net over F2, using
(11, 11+9, 48)-Net over F2 — Upper bound on s (digital)
There is no digital (11, 20, 49)-net over F2, because
- 1 times m-reduction [i] would yield digital (11, 19, 49)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(219, 49, F2, 8) (dual of [49, 30, 9]-code), but
- construction Y1 [i] would yield
- linear OA(218, 35, F2, 8) (dual of [35, 17, 9]-code), but
- linear OA(230, 49, F2, 14) (dual of [49, 19, 15]-code), but
- discarding factors / shortening the dual code would yield linear OA(230, 48, F2, 14) (dual of [48, 18, 15]-code), but
- adding a parity check bit [i] would yield linear OA(231, 49, F2, 15) (dual of [49, 18, 16]-code), but
- discarding factors / shortening the dual code would yield linear OA(230, 48, F2, 14) (dual of [48, 18, 15]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(219, 49, F2, 8) (dual of [49, 30, 9]-code), but
(11, 11+9, 54)-Net in Base 2 — Upper bound on s
There is no (11, 20, 55)-net in base 2, because
- 1 times m-reduction [i] would yield (11, 19, 55)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 550771 > 219 [i]