MinT - Best Known (167−61, 167, s)-Nets in Base 2

Best Known (167−61, 167, s)-Nets in Base 2

(167−61, 167, 68)-Net over F₂ — Constructive and digital

Digital (106, 167, 68)-net over F₂, using

3 times m-reduction [i] based on digital (106, 170, 68)-net over F₂, using
- trace code for nets [i] based on digital (21, 85, 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]

(167−61, 167, 98)-Net over F₂ — Digital

Digital (106, 167, 98)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(167−61, 167, 514)-Net in Base 2 — Upper bound on s

There is no (106, 167, 515)-net in base 2, because

1 times m-reduction [i] would yield (106, 166, 515)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 94 212151 579011 793912 760936 376132 235283 369747 587968 > 2¹⁶⁶ [i]