MinT - Best Known (82−23, 82, s)-Nets in Base 2

Best Known (82−23, 82, s)-Nets in Base 2

(82−23, 82, 75)-Net over F₂ — Constructive and digital

Digital (59, 82, 75)-net over F₂, using

2¹ times duplication [i] based on digital (58, 81, 75)-net over F₂, using
- trace code for nets [i] based on digital (4, 27, 25)-net over F₈, using
  - net from sequence [i] based on digital (4, 24)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 4 and N(F) ≥ 25, using
      - a function field by SÃ©mirat [i]

(82−23, 82, 105)-Net over F₂ — Digital

Digital (59, 82, 105)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(82−23, 82, 792)-Net in Base 2 — Upper bound on s

There is no (59, 82, 793)-net in base 2, because

1 times m-reduction [i] would yield (59, 81, 793)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 2 429109 923591 246487 518616 > 2⁸¹ [i]