MinT - Best Known (225−83, 225, s)-Nets in Base 2

Best Known (225−83, 225, s)-Nets in Base 2

(225−83, 225, 76)-Net over F₂ — Constructive and digital

Digital (142, 225, 76)-net over F₂, using

3 times m-reduction [i] based on digital (142, 228, 76)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (45, 88, 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]
  2. digital (54, 140, 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]

(225−83, 225, 84)-Net in Base 2 — Constructive

(142, 225, 84)-net in base 2, using

5 times m-reduction [i] based on (142, 230, 84)-net in base 2, using
- trace code for nets [i] based on (27, 115, 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]

(225−83, 225, 123)-Net over F₂ — Digital

Digital (142, 225, 123)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(225−83, 225, 653)-Net in Base 2 — Upper bound on s

There is no (142, 225, 654)-net in base 2, because

1 times m-reduction [i] would yield (142, 224, 654)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 27 871701 284252 024046 111071 436470 962261 822825 017560 337372 142185 395580 > 2²²⁴ [i]