MinT - Best Known (169−53, 169, s)-Nets in Base 2

Best Known (169−53, 169, s)-Nets in Base 2

(169−53, 169, 76)-Net over F₂ — Constructive and digital

Digital (116, 169, 76)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (36, 62, 34)-net over F₂, using
  - trace code for nets [i] based on digital (5, 31, 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 (54, 107, 42)-net over F₂, using
  - net from sequence [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]

(169−53, 169, 86)-Net in Base 2 — Constructive

(116, 169, 86)-net in base 2, using

3 times m-reduction [i] based on (116, 172, 86)-net in base 2, using
- trace code for nets [i] based on (30, 86, 43)-net in base 4, using
  - net from sequence [i] based on (30, 42)-sequence in base 4, using
    - base expansion [i] based on digital (60, 42)-sequence over F₂, using
      - t-expansion [i] based on digital (59, 42)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

(169−53, 169, 140)-Net over F₂ — Digital

Digital (116, 169, 140)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(169−53, 169, 892)-Net in Base 2 — Upper bound on s

There is no (116, 169, 893)-net in base 2, because

1 times m-reduction [i] would yield (116, 168, 893)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 384 158305 473831 879099 332177 935119 435309 732656 644778 > 2¹⁶⁸ [i]