MinT - Best Known (42, 65, s)-Nets in Base 2

Best Known (42, 65, s)-Nets in Base 2

(42, 65, 44)-Net over F₂ — Constructive and digital

Digital (42, 65, 44)-net over F₂, using

1 times m-reduction [i] based on digital (42, 66, 44)-net over F₂, using
- trace code for nets [i] based on digital (9, 33, 22)-net over F₄, using
  - net from sequence [i] based on digital (9, 21)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 9 and N(F) ≥ 22, using
      - an explicitly constructive algebraic function field [i]

(42, 65, 52)-Net over F₂ — Digital

Digital (42, 65, 52)-net over F₂, using

1 times m-reduction [i] based on digital (42, 66, 52)-net over F₂, using
- trace code for nets [i] based on digital (9, 33, 26)-net over F₄, using
  - net from sequence [i] based on digital (9, 25)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 9 and N(F) ≥ 26, using
      - Drinfeld modules of rank 1 [i]

(42, 65, 261)-Net in Base 2 — Upper bound on s

There is no (42, 65, 262)-net in base 2, because

1 times m-reduction [i] would yield (42, 64, 262)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 18 898327 568102 292544 > 2⁶⁴ [i]