MinT - Best Known (77, 77+31, s)-Nets in Base 2

Best Known (77, 77+31, s)-Nets in Base 2

(77, 77+31, 84)-Net over F₂ — Constructive and digital

Digital (77, 108, 84)-net over F₂, using

trace code for nets [i] based on digital (5, 36, 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]

(77, 77+31, 122)-Net over F₂ — Digital

Digital (77, 108, 122)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(77, 77+31, 880)-Net in Base 2 — Upper bound on s

There is no (77, 108, 881)-net in base 2, because

1 times m-reduction [i] would yield (77, 107, 881)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 164 849371 094991 373821 553000 611840 > 2¹⁰⁷ [i]