MinT - Best Known (147−45, 147, s)-Nets in Base 2

Best Known (147−45, 147, s)-Nets in Base 2

(147−45, 147, 75)-Net over F₂ — Constructive and digital

Digital (102, 147, 75)-net over F₂, using

trace code for nets [i] based on digital (4, 49, 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]

(147−45, 147, 84)-Net in Base 2 — Constructive

(102, 147, 84)-net in base 2, using

3 times m-reduction [i] based on (102, 150, 84)-net in base 2, using
- trace code for nets [i] based on (27, 75, 42)-net in base 4, using
  - net from sequence [i] based on (27, 41)-sequence in base 4, using
    - base expansion [i] based on digital (54, 41)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
        an explicitly constructive algebraic function field [i]

(147−45, 147, 132)-Net over F₂ — Digital

Digital (102, 147, 132)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(147−45, 147, 868)-Net in Base 2 — Upper bound on s

There is no (102, 147, 869)-net in base 2, because

1 times m-reduction [i] would yield (102, 146, 869)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 89 726979 055536 369428 976703 222594 527710 664168 > 2¹⁴⁶ [i]