MinT - Best Known (111, 111+37, s)-Nets in Base 2

Best Known (111, 111+37, s)-Nets in Base 2

(111, 111+37, 144)-Net over F₂ — Constructive and digital

Digital (111, 148, 144)-net over F₂, using

2 times m-reduction [i] based on digital (111, 150, 144)-net over F₂, using
- trace code for nets [i] based on digital (11, 50, 48)-net over F₈, using
  - net from sequence [i] based on digital (11, 47)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 11 and N(F) ≥ 48, using
      - a function field by SÃ©mirat [i]

(111, 111+37, 219)-Net over F₂ — Digital

Digital (111, 148, 219)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(111, 111+37, 2144)-Net in Base 2 — Upper bound on s

There is no (111, 148, 2145)-net in base 2, because

1 times m-reduction [i] would yield (111, 147, 2145)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 179 808419 641203 070387 328111 132766 093101 604728 > 2¹⁴⁷ [i]