MinT - Best Known (189−65, 189, s)-Nets in Base 2

Best Known (189−65, 189, s)-Nets in Base 2

(189−65, 189, 70)-Net over F₂ — Constructive and digital

Digital (124, 189, 70)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (38, 70, 28)-net over F₂, using
  - trace code for nets [i] based on digital (3, 35, 14)-net over F₄, using
    - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
        an explicitly constructive algebraic function field [i]
2. digital (54, 119, 42)-net over F₂, using
  - net from sequence [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]

(189−65, 189, 84)-Net in Base 2 — Constructive

(124, 189, 84)-net in base 2, using

5 times m-reduction [i] based on (124, 194, 84)-net in base 2, using
- trace code for nets [i] based on (27, 97, 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]

(189−65, 189, 124)-Net over F₂ — Digital

Digital (124, 189, 124)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(189−65, 189, 704)-Net in Base 2 — Upper bound on s

There is no (124, 189, 705)-net in base 2, because

1 times m-reduction [i] would yield (124, 188, 705)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 401 322490 667950 540275 537467 784150 387368 429564 300311 469810 > 2¹⁸⁸ [i]