MinT - Best Known (244−57, 244, s)-Nets in Base 2

Best Known (244−57, 244, s)-Nets in Base 2

(244−57, 244, 202)-Net over F₂ — Constructive and digital

Digital (187, 244, 202)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 31, 7)-net over F₂, using
  - net from sequence [i] based on digital (3, 6)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 3 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (156, 213, 195)-net over F₂, using
  - trace code for nets [i] based on digital (14, 71, 65)-net over F₈, using
    - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
        the Suzuki function field over F₈ [i]

(244−57, 244, 397)-Net over F₂ — Digital

Digital (187, 244, 397)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(244−57, 244, 4587)-Net in Base 2 — Upper bound on s

There is no (187, 244, 4588)-net in base 2, because

1 times m-reduction [i] would yield (187, 243, 4588)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 14 137368 124781 174712 294645 923613 829841 559632 051207 722061 256770 159900 660520 > 2²⁴³ [i]