MinT - Best Known (9−4, 9, s)-Nets in Base 2

Best Known (9−4, 9, s)-Nets in Base 2

(9−4, 9, 23)-Net over F₂ — Constructive and digital

Digital (5, 9, 23)-net over F₂, using

linear code embedding found in a computer search [i]

(9−4, 9, 23)-Net over F₂ — Upper bound on s (digital)

There is no digital (5, 9, 24)-net over F₂, because

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]

(9−4, 9, 29)-Net in Base 2 — Upper bound on s

There is no (5, 9, 30)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 526 > 2⁹ [i]
extracting embedded orthogonal array [i] would yield OA(2⁹, 30, S₂, 4), but
- the linear programming bound shows that MÂ â‰¥ 108864 / 209 > 2⁹ [i]