MinT - Best Known (259−105, 259, s)-Nets in Base 2

Best Known (259−105, 259, s)-Nets in Base 2

(259−105, 259, 77)-Net over F₂ — Constructive and digital

Digital (154, 259, 77)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (48, 100, 35)-net over F₂, using
  - net from sequence [i] based on digital (48, 34)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 2 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
2. digital (54, 159, 42)-net over F₂, using
  - net from sequence [i] based on digital (54, 41)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
      - an explicitly constructive algebraic function field [i]

(259−105, 259, 112)-Net over F₂ — Digital

Digital (154, 259, 112)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(259−105, 259, 556)-Net in Base 2 — Upper bound on s

There is no (154, 259, 557)-net in base 2, because

1 times m-reduction [i] would yield (154, 258, 557)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 471843 836671 750259 757661 293590 395870 955680 694795 741407 286632 638192 329920 339104 > 2²⁵⁸ [i]