MinT - Best Known (167−51, 167, s)-Nets in Base 2

Best Known (167−51, 167, s)-Nets in Base 2

(167−51, 167, 78)-Net over F₂ — Constructive and digital

Digital (116, 167, 78)-net over F₂, using

2¹ times duplication [i] based on digital (115, 166, 78)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 34, 12)-net over F₂, using
    - net from sequence [i] based on digital (9, 11)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 9 and N(F) ≥ 12, using
        an explicitly constructive algebraic function field [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]

(167−51, 167, 86)-Net in Base 2 — Constructive

(116, 167, 86)-net in base 2, using

5 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]

(167−51, 167, 148)-Net over F₂ — Digital

Digital (116, 167, 148)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(167−51, 167, 978)-Net in Base 2 — Upper bound on s

There is no (116, 167, 979)-net in base 2, because

1 times m-reduction [i] would yield (116, 166, 979)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 94 459344 987303 276472 457737 069217 606493 483914 436896 > 2¹⁶⁶ [i]