MinT - Best Known (159−51, 159, s)-Nets in Base 2

Best Known (159−51, 159, s)-Nets in Base 2

(159−51, 159, 72)-Net over F₂ — Constructive and digital

Digital (108, 159, 72)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (2, 27, 6)-net over F₂, using
  - net from sequence [i] based on digital (2, 5)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 2 and N(F) ≥ 6, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (81, 132, 66)-net over F₂, using
  - trace code for nets [i] based on digital (15, 66, 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]

(159−51, 159, 84)-Net in Base 2 — Constructive

(108, 159, 84)-net in base 2, using

3 times m-reduction [i] based on (108, 162, 84)-net in base 2, using
- trace code for nets [i] based on (27, 81, 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]

(159−51, 159, 126)-Net over F₂ — Digital

Digital (108, 159, 126)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(159−51, 159, 776)-Net in Base 2 — Upper bound on s

There is no (108, 159, 777)-net in base 2, because

1 times m-reduction [i] would yield (108, 158, 777)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 367247 668406 835791 186876 939612 927806 932167 944288 > 2¹⁵⁸ [i]