MinT - Best Known (109, 161, s)-Nets in Base 2

Best Known (109, 161, s)-Nets in Base 2

(109, 161, 71)-Net over F₂ — Constructive and digital

Digital (109, 161, 71)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 27, 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 (82, 134, 66)-net over F₂, using
  - trace code for nets [i] based on digital (15, 67, 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]

(109, 161, 84)-Net in Base 2 — Constructive

(109, 161, 84)-net in base 2, using

3 times m-reduction [i] based on (109, 164, 84)-net in base 2, using
- trace code for nets [i] based on (27, 82, 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]

(109, 161, 125)-Net over F₂ — Digital

Digital (109, 161, 125)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(109, 161, 733)-Net in Base 2 — Upper bound on s

There is no (109, 161, 734)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 2 932609 466734 017876 747822 399651 141170 709153 209110 > 2¹⁶¹ [i]