MinT - Best Known (225−85, 225, s)-Nets in Base 2

Best Known (225−85, 225, s)-Nets in Base 2

(225−85, 225, 76)-Net over F₂ — Constructive and digital

Digital (140, 225, 76)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (39, 81, 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 (59, 144, 43)-net over F₂, using
  - net from sequence [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]

(225−85, 225, 84)-Net in Base 2 — Constructive

(140, 225, 84)-net in base 2, using

1 times m-reduction [i] based on (140, 226, 84)-net in base 2, using
- trace code for nets [i] based on (27, 113, 42)-net in base 4, using
  - net from sequence [i] based on (27, 41)-sequence in base 4, using
    - base expansion [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]

(225−85, 225, 116)-Net over F₂ — Digital

Digital (140, 225, 116)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(225−85, 225, 605)-Net in Base 2 — Upper bound on s

There is no (140, 225, 606)-net in base 2, because

1 times m-reduction [i] would yield (140, 224, 606)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 27 337202 465474 568944 954594 364538 891456 006775 684831 174814 703753 847700 > 2²²⁴ [i]