MinT - Best Known (232−145, 232, s)-Nets in Base 2

Best Known (232−145, 232, s)-Nets in Base 2

(232−145, 232, 52)-Net over F₂ — Constructive and digital

Digital (87, 232, 52)-net over F₂, using

t-expansion [i] based on digital (85, 232, 52)-net over F₂, using
- net from sequence [i] based on digital (85, 51)-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 3 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]

(232−145, 232, 57)-Net over F₂ — Digital

Digital (87, 232, 57)-net over F₂, using

t-expansion [i] based on digital (83, 232, 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]

(232−145, 232, 164)-Net in Base 2 — Upper bound on s

There is no (87, 232, 165)-net in base 2, because

1 times m-reduction [i] would yield (87, 231, 165)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3547 073574 254575 535284 328064 453979 069314 919841 659775 938331 707816 673272 > 2²³¹ [i]