MinT - Best Known (177−59, 177, s)-Nets in Base 2

Best Known (177−59, 177, s)-Nets in Base 2

(177−59, 177, 70)-Net over F₂ — Constructive and digital

Digital (118, 177, 70)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (35, 64, 28)-net over F₂, using
  - trace code for nets [i] based on digital (3, 32, 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, 113, 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]

(177−59, 177, 84)-Net in Base 2 — Constructive

(118, 177, 84)-net in base 2, using

5 times m-reduction [i] based on (118, 182, 84)-net in base 2, using
- trace code for nets [i] based on (27, 91, 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]

(177−59, 177, 126)-Net over F₂ — Digital

Digital (118, 177, 126)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(177−59, 177, 741)-Net in Base 2 — Upper bound on s

There is no (118, 177, 742)-net in base 2, because

1 times m-reduction [i] would yield (118, 176, 742)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 97247 499088 315393 670300 521678 321506 688876 539696 518708 > 2¹⁷⁶ [i]