MinT - Best Known (64−23, 64, s)-Nets in Base 2

Best Known (64−23, 64, s)-Nets in Base 2

(64−23, 64, 44)-Net over F₂ — Constructive and digital

Digital (41, 64, 44)-net over F₂, using

trace code for nets [i] based on digital (9, 32, 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]

(64−23, 64, 52)-Net over F₂ — Digital

Digital (41, 64, 52)-net over F₂, using

trace code for nets [i] based on digital (9, 32, 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]

(64−23, 64, 244)-Net in Base 2 — Upper bound on s

There is no (41, 64, 245)-net in base 2, because

1 times m-reduction [i] would yield (41, 63, 245)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 9 420171 201573 289984 > 2⁶³ [i]