MinT - Best Known (225−81, 225, s)-Nets in Base 2

Best Known (225−81, 225, s)-Nets in Base 2

(225−81, 225, 77)-Net over F₂ — Constructive and digital

Digital (144, 225, 77)-net over F₂, using

3 times m-reduction [i] based on digital (144, 228, 77)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (48, 90, 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, 138, 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]

(225−81, 225, 86)-Net in Base 2 — Constructive

(144, 225, 86)-net in base 2, using

3 times m-reduction [i] based on (144, 228, 86)-net in base 2, using
- trace code for nets [i] based on (30, 114, 43)-net in base 4, using
  - net from sequence [i] based on (30, 42)-sequence in base 4, using
    - base expansion [i] based on digital (60, 42)-sequence over F₂, using
      - t-expansion [i] based on digital (59, 42)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

(225−81, 225, 129)-Net over F₂ — Digital

Digital (144, 225, 129)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(225−81, 225, 707)-Net in Base 2 — Upper bound on s

There is no (144, 225, 708)-net in base 2, because

1 times m-reduction [i] would yield (144, 224, 708)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 27 890390 367090 391838 361986 502299 199557 214354 598406 867560 613117 661720 > 2²²⁴ [i]