MinT - Best Known (227−143, 227, s)-Nets in Base 2

Best Known (227−143, 227, s)-Nets in Base 2

(227−143, 227, 51)-Net over F₂ — Constructive and digital

Digital (84, 227, 51)-net over F₂, using

t-expansion [i] based on digital (80, 227, 51)-net over F₂, using
- net from sequence [i] based on digital (80, 50)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, 1 place with degree 2, and 2 places with degree 6 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(227−143, 227, 57)-Net over F₂ — Digital

Digital (84, 227, 57)-net over F₂, using

t-expansion [i] based on digital (83, 227, 57)-net over F₂, using
- net from sequence [i] based on digital (83, 56)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 83 and N(F) ≥ 57, using
    - an algebraic function field by Auer [i]

(227−143, 227, 158)-Net in Base 2 — Upper bound on s

There is no (84, 227, 159)-net in base 2, because

1 times m-reduction [i] would yield (84, 226, 159)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 121 151620 961510 799654 962678 571282 104969 029264 350159 155543 575754 300780 > 2²²⁶ [i]