MinT - Best Known (169−55, 169, s)-Nets in Base 2

Best Known (169−55, 169, s)-Nets in Base 2

(169−55, 169, 72)-Net over F₂ — Constructive and digital

Digital (114, 169, 72)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (2, 29, 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 (85, 140, 66)-net over F₂, using
  - trace code for nets [i] based on digital (15, 70, 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]

(169−55, 169, 84)-Net in Base 2 — Constructive

(114, 169, 84)-net in base 2, using

5 times m-reduction [i] based on (114, 174, 84)-net in base 2, using
- trace code for nets [i] based on (27, 87, 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]

(169−55, 169, 128)-Net over F₂ — Digital

Digital (114, 169, 128)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(169−55, 169, 776)-Net in Base 2 — Upper bound on s

There is no (114, 169, 777)-net in base 2, because

1 times m-reduction [i] would yield (114, 168, 777)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 380 616716 058559 072696 130564 246759 194005 599678 891904 > 2¹⁶⁸ [i]