MinT - Best Known (216−125, 216, s)-Nets in Base 2

Best Known (216−125, 216, s)-Nets in Base 2

(216−125, 216, 53)-Net over F₂ — Constructive and digital

Digital (91, 216, 53)-net over F₂, using

t-expansion [i] based on digital (90, 216, 53)-net over F₂, using
- net from sequence [i] based on digital (90, 52)-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 4 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]

(216−125, 216, 57)-Net over F₂ — Digital

Digital (91, 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]

(216−125, 216, 184)-Net in Base 2 — Upper bound on s

There is no (91, 216, 185)-net in base 2, because

1 times m-reduction [i] would yield (91, 215, 185)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 55817 833901 710325 202029 761577 761671 364695 007318 669292 391258 532864 > 2²¹⁵ [i]