MinT - Best Known (216−129, 216, s)-Nets in Base 2

Best Known (216−129, 216, s)-Nets in Base 2

(216−129, 216, 52)-Net over F₂ — Constructive and digital

Digital (87, 216, 52)-net over F₂, using

t-expansion [i] based on digital (85, 216, 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]

(216−129, 216, 57)-Net over F₂ — Digital

Digital (87, 216, 57)-net over F₂, using

t-expansion [i] based on digital (83, 216, 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]

(216−129, 216, 171)-Net in Base 2 — Upper bound on s

There is no (87, 216, 172)-net in base 2, because

1 times m-reduction [i] would yield (87, 215, 172)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 60493 634967 989141 979977 419881 810551 267955 026885 635774 596370 219074 > 2²¹⁵ [i]