MinT - Best Known (89, 89+53, s)-Nets in Base 2

Best Known (89, 89+53, s)-Nets in Base 2

(89, 89+53, 66)-Net over F₂ — Constructive and digital

Digital (89, 142, 66)-net over F₂, using

6 times m-reduction [i] based on digital (89, 148, 66)-net over F₂, using
- trace code for nets [i] based on digital (15, 74, 33)-net over F₄, using
  - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]

(89, 89+53, 82)-Net over F₂ — Digital

Digital (89, 142, 82)-net over F₂, using

trace code for nets [i] based on digital (18, 71, 41)-net over F₄, using
- net from sequence [i] based on digital (18, 40)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 18 and N(F) ≥ 41, using
    - Drinfeld modules of rank 1 [i]

(89, 89+53, 415)-Net in Base 2 — Upper bound on s

There is no (89, 142, 416)-net in base 2, because

1 times m-reduction [i] would yield (89, 141, 416)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 2 849471 562930 877802 120046 225961 380529 788991 > 2¹⁴¹ [i]