MinT - Best Known (167−57, 167, s)-Nets in Base 2

Best Known (167−57, 167, s)-Nets in Base 2

(167−57, 167, 68)-Net over F₂ — Constructive and digital

Digital (110, 167, 68)-net over F₂, using

11 times m-reduction [i] based on digital (110, 178, 68)-net over F₂, using
- trace code for nets [i] based on digital (21, 89, 34)-net over F₄, using
  - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
      - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(167−57, 167, 72)-Net in Base 2 — Constructive

(110, 167, 72)-net in base 2, using

1 times m-reduction [i] based on (110, 168, 72)-net in base 2, using
- trace code for nets [i] based on (26, 84, 36)-net in base 4, using
  - net from sequence [i] based on (26, 35)-sequence in base 4, using
    - base expansion [i] based on digital (52, 35)-sequence over F₂, using
      - t-expansion [i] based on digital (51, 35)-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 3 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]

(167−57, 167, 114)-Net over F₂ — Digital

Digital (110, 167, 114)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(167−57, 167, 647)-Net in Base 2 — Upper bound on s

There is no (110, 167, 648)-net in base 2, because

1 times m-reduction [i] would yield (110, 166, 648)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 94 468957 667000 992161 270596 538113 413674 945825 391696 > 2¹⁶⁶ [i]