MinT - Best Known (84, 84+129, s)-Nets in Base 2

Best Known (84, 84+129, s)-Nets in Base 2

(84, 84+129, 51)-Net over F₂ — Constructive and digital

Digital (84, 213, 51)-net over F₂, using

t-expansion [i] based on digital (80, 213, 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]

(84, 84+129, 57)-Net over F₂ — Digital

Digital (84, 213, 57)-net over F₂, using

t-expansion [i] based on digital (83, 213, 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]

(84, 84+129, 163)-Net in Base 2 — Upper bound on s

There is no (84, 213, 164)-net in base 2, because

1 times m-reduction [i] would yield (84, 212, 164)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 7025 748013 787220 972996 729772 090803 073731 068370 833834 432147 595858 > 2²¹² [i]