MinT - Best Known (254−111, 254, s)-Nets in Base 2

Best Known (254−111, 254, s)-Nets in Base 2

(254−111, 254, 68)-Net over F₂ — Constructive and digital

Digital (143, 254, 68)-net over F₂, using

1 times m-reduction [i] based on digital (143, 255, 68)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 95, 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 (48, 160, 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]

(254−111, 254, 92)-Net over F₂ — Digital

Digital (143, 254, 92)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(254−111, 254, 440)-Net in Base 2 — Upper bound on s

There is no (143, 254, 441)-net in base 2, because

1 times m-reduction [i] would yield (143, 253, 441)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 14538 463223 647633 935866 562917 038142 689277 818057 437676 805937 813289 521639 645440 > 2²⁵³ [i]