MinT - Best Known (81, 81+31, s)-Nets in Base 2

Best Known (81, 81+31, s)-Nets in Base 2

(81, 81+31, 84)-Net over F₂ — Constructive and digital

Digital (81, 112, 84)-net over F₂, using

2 times m-reduction [i] based on digital (81, 114, 84)-net over F₂, using
- trace code for nets [i] based on digital (5, 38, 28)-net over F₈, using
  - net from sequence [i] based on digital (5, 27)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
      - a function field by SÃ©mirat [i]

(81, 81+31, 137)-Net over F₂ — Digital

Digital (81, 112, 137)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(81, 81+31, 1063)-Net in Base 2 — Upper bound on s

There is no (81, 112, 1064)-net in base 2, because

1 times m-reduction [i] would yield (81, 111, 1064)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 2628 751756 058053 939395 207741 714348 > 2¹¹¹ [i]