MinT - Best Known (81−31, 81, s)-Nets in Base 2

Best Known (81−31, 81, s)-Nets in Base 2

(81−31, 81, 44)-Net over F₂ — Constructive and digital

Digital (50, 81, 44)-net over F₂, using

1 times m-reduction [i] based on digital (50, 82, 44)-net over F₂, using
- trace code for nets [i] based on digital (9, 41, 22)-net over F₄, using
  - net from sequence [i] based on digital (9, 21)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 9 and N(F) ≥ 22, using
      - an explicitly constructive algebraic function field [i]

(81−31, 81, 52)-Net over F₂ — Digital

Digital (50, 81, 52)-net over F₂, using

1 times m-reduction [i] based on digital (50, 82, 52)-net over F₂, using
- trace code for nets [i] based on digital (9, 41, 26)-net over F₄, using
  - net from sequence [i] based on digital (9, 25)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 9 and N(F) ≥ 26, using
      - Drinfeld modules of rank 1 [i]

(81−31, 81, 237)-Net in Base 2 — Upper bound on s

There is no (50, 81, 238)-net in base 2, because

1 times m-reduction [i] would yield (50, 80, 238)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1 224083 489080 943709 929948 > 2⁸⁰ [i]