MinT - Best Known (5, 5+5, s)-Nets in Base 2

Best Known (5, 5+5, s)-Nets in Base 2

(5, 5+5, 20)-Net over F₂ — Constructive and digital

Digital (5, 10, 20)-net over F₂, using

linear code embedding found in a computer search [i]

(5, 5+5, 23)-Net over F₂ — Upper bound on s (digital)

There is no digital (5, 10, 24)-net over F₂, because

1 times m-reduction [i] would yield digital (5, 9, 24)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁹, 24, F₂, 4) (dual of [24, 15, 5]-code), but
  - adding a parity check bit [i] would yield linear OA(2¹⁰, 25, F₂, 5) (dual of [25, 15, 6]-code), but
    - bound by Simonis [i]

(5, 5+5, 26)-Net in Base 2 — Upper bound on s

There is no (5, 10, 27)-net in base 2, because

extracting embedded OOA [i] would yield OOA(2¹⁰, 27, S₂, 2, 5), but
- the linear programming bound for OOAs shows that MÂ â‰¥ 523 232769 812480 / 490066 468849 > 2¹⁰ [i]