MinT - Best Known (81, 81+37, s)-Nets in Base 2

Best Known (81, 81+37, s)-Nets in Base 2

(81, 81+37, 68)-Net over F₂ — Constructive and digital

Digital (81, 118, 68)-net over F₂, using

2 times m-reduction [i] based on digital (81, 120, 68)-net over F₂, using
- trace code for nets [i] based on digital (21, 60, 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]

(81, 81+37, 105)-Net over F₂ — Digital

Digital (81, 118, 105)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(81, 81+37, 657)-Net in Base 2 — Upper bound on s

There is no (81, 118, 658)-net in base 2, because

1 times m-reduction [i] would yield (81, 117, 658)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 167942 689533 873855 634564 433152 469960 > 2¹¹⁷ [i]