MinT - Best Known (86, 86+144, s)-Nets in Base 2

Best Known (86, 86+144, s)-Nets in Base 2

(86, 86+144, 52)-Net over F₂ — Constructive and digital

Digital (86, 230, 52)-net over F₂, using

t-expansion [i] based on digital (85, 230, 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]

(86, 86+144, 57)-Net over F₂ — Digital

Digital (86, 230, 57)-net over F₂, using

t-expansion [i] based on digital (83, 230, 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]

(86, 86+144, 162)-Net in Base 2 — Upper bound on s

There is no (86, 230, 163)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1948 135434 592509 476261 184071 832495 626405 483968 506075 782736 114749 013256 > 2²³⁰ [i]