MinT - Best Known (3, 8, s)-Nets in Base 2

Best Known (3, 8, s)-Nets in Base 2

(3, 8, 10)-Net over F₂ — Constructive and digital

Digital (3, 8, 10)-net over F₂, using

linear code embedding found in a computer search [i]

(3, 8, 11)-Net over F₂ — Upper bound on s (digital)

There is no digital (3, 8, 12)-net over F₂, because

1 times m-reduction [i] would yield digital (3, 7, 12)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁷, 12, F₂, 4) (dual of [12, 5, 5]-code), but
  - bound by Fontaine and Peterson [i]

(3, 8, 12)-Net in Base 2 — Upper bound on s

There is no (3, 8, 13)-net in base 2, because

extracting embedded OOA [i] would yield OOA(2⁸, 13, S₂, 2, 5), but
- the linear programming bound for OOAs shows that MÂ â‰¥ 23 575424 / 81053 > 2⁸ [i]