MinT - Best Known (167−55, 167, s)-Nets in Base 2

Best Known (167−55, 167, s)-Nets in Base 2

(167−55, 167, 69)-Net over F₂ — Constructive and digital

Digital (112, 167, 69)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (37, 64, 34)-net over F₂, using
  - trace code for nets [i] based on digital (5, 32, 17)-net over F₄, using
    - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
        an explicitly constructive algebraic function field [i]
2. digital (48, 103, 35)-net over F₂, using
  - net from sequence [i] based on digital (48, 34)-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 2 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]

(167−55, 167, 84)-Net in Base 2 — Constructive

(112, 167, 84)-net in base 2, using

3 times m-reduction [i] based on (112, 170, 84)-net in base 2, using
- trace code for nets [i] based on (27, 85, 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]

(167−55, 167, 123)-Net over F₂ — Digital

Digital (112, 167, 123)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(167−55, 167, 735)-Net in Base 2 — Upper bound on s

There is no (112, 167, 736)-net in base 2, because

1 times m-reduction [i] would yield (112, 166, 736)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 94 497021 596791 245702 537093 628828 922807 528007 153659 > 2¹⁶⁶ [i]