MinT - Best Known (143−45, 143, s)-Nets in Base 2

Best Known (143−45, 143, s)-Nets in Base 2

(143−45, 143, 71)-Net over F₂ — Constructive and digital

Digital (98, 143, 71)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 23, 5)-net over F₂, using
  - net from sequence [i] based on digital (1, 4)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 1 and N(F) ≥ 5, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (75, 120, 66)-net over F₂, using
  - trace code for nets [i] based on digital (15, 60, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]

(143−45, 143, 72)-Net in Base 2 — Constructive

(98, 143, 72)-net in base 2, using

1 times m-reduction [i] based on (98, 144, 72)-net in base 2, using
- trace code for nets [i] based on (26, 72, 36)-net in base 4, using
  - net from sequence [i] based on (26, 35)-sequence in base 4, using
    - base expansion [i] based on digital (52, 35)-sequence over F₂, using
      - t-expansion [i] based on digital (51, 35)-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 3 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]

(143−45, 143, 122)-Net over F₂ — Digital

Digital (98, 143, 122)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(143−45, 143, 762)-Net in Base 2 — Upper bound on s

There is no (98, 143, 763)-net in base 2, because

1 times m-reduction [i] would yield (98, 142, 763)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 5 704256 384858 969737 698632 123889 619802 008082 > 2¹⁴² [i]