MinT - Best Known (260−117, 260, s)-Nets in Base 2

Best Known (260−117, 260, s)-Nets in Base 2

(260−117, 260, 67)-Net over F₂ — Constructive and digital

Digital (143, 260, 67)-net over F₂, using

2¹ times duplication [i] based on digital (142, 259, 67)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 97, 33)-net over F₂, using
    - net from sequence [i] based on digital (39, 32)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 39 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (45, 162, 34)-net over F₂, using
    - net from sequence [i] based on digital (45, 33)-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 1 place 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]

(260−117, 260, 88)-Net over F₂ — Digital

Digital (143, 260, 88)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(260−117, 260, 415)-Net in Base 2 — Upper bound on s

There is no (143, 260, 416)-net in base 2, because

1 times m-reduction [i] would yield (143, 259, 416)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 947647 015019 621710 176206 462382 521298 857854 150655 702760 133848 119370 157077 805860 > 2²⁵⁹ [i]