MinT - Best Known (42, 85, s)-Nets in Base 2

Best Known (42, 85, s)-Nets in Base 2

(42, 85, 33)-Net over F₂ — Constructive and digital

Digital (42, 85, 33)-net over F₂, using

t-expansion [i] based on digital (39, 85, 33)-net over F₂, using
- net from sequence [i] based on digital (39, 32)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 39 and N(F) ≥ 33, using
    - an explicitly constructive algebraic function field [i]

(42, 85, 94)-Net over F₂ — Upper bound on s (digital)

There is no digital (42, 85, 95)-net over F₂, because

1 times m-reduction [i] would yield digital (42, 84, 95)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁸⁴, 95, F₂, 42) (dual of [95, 11, 43]-code), but
  - adding a parity check bit [i] would yield linear OA(2⁸⁵, 96, F₂, 43) (dual of [96, 11, 44]-code), but
    - â€œBroâ€ bound on codes from Brouwerâ€™s database [i]

(42, 85, 97)-Net in Base 2 — Upper bound on s

There is no (42, 85, 98)-net in base 2, because

1 times m-reduction [i] would yield (42, 84, 98)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁸⁴, 98, S₂, 42), but
  - the linear programming bound shows that MÂ â‰¥ 28936 848418 295763 925767 028736 / 1443 > 2⁸⁴ [i]