MinT - Best Known (31, 31+19, s)-Nets in Base 2

Best Known (31, 31+19, s)-Nets in Base 2

(31, 31+19, 34)-Net over F₂ — Constructive and digital

Digital (31, 50, 34)-net over F₂, using

2 times m-reduction [i] based on digital (31, 52, 34)-net over F₂, using
- trace code for nets [i] based on digital (5, 26, 17)-net over F₄, using
  - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]

(31, 31+19, 40)-Net over F₂ — Digital

Digital (31, 50, 40)-net over F₂, using

trace code for nets [i] based on digital (6, 25, 20)-net over F₄, using
- net from sequence [i] based on digital (6, 19)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 6 and N(F) ≥ 20, using
    - Drinfeld modules of rank 1 [i]

(31, 31+19, 167)-Net in Base 2 — Upper bound on s

There is no (31, 50, 168)-net in base 2, because

1 times m-reduction [i] would yield (31, 49, 168)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 563 172843 165860 > 2⁴⁹ [i]