MinT - Best Known (231−129, 231, s)-Nets in Base 2

Best Known (231−129, 231, s)-Nets in Base 2

(231−129, 231, 55)-Net over F₂ — Constructive and digital

Digital (102, 231, 55)-net over F₂, using

t-expansion [i] based on digital (100, 231, 55)-net over F₂, using
- net from sequence [i] based on digital (100, 54)-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 6 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]

(231−129, 231, 65)-Net over F₂ — Digital

Digital (102, 231, 65)-net over F₂, using

t-expansion [i] based on digital (95, 231, 65)-net over F₂, using
- net from sequence [i] based on digital (95, 64)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 95 and N(F) ≥ 65, using
    - Drinfeld modules of rank 1 [i]

(231−129, 231, 214)-Net in Base 2 — Upper bound on s

There is no (102, 231, 215)-net in base 2, because

1 times m-reduction [i] would yield (102, 230, 215)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1987 360233 016486 710134 390827 044902 186645 361366 873559 192569 736737 819027 > 2²³⁰ [i]