MinT - Best Known (127, 127+69, s)-Nets in Base 2

Best Known (127, 127+69, s)-Nets in Base 2

(127, 127+69, 75)-Net over F₂ — Constructive and digital

Digital (127, 196, 75)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (39, 73, 33)-net over F₂, using
  - net from sequence [i] based on digital (39, 32)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 39 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (54, 123, 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]

(127, 127+69, 84)-Net in Base 2 — Constructive

(127, 196, 84)-net in base 2, using

4 times m-reduction [i] based on (127, 200, 84)-net in base 2, using
- trace code for nets [i] based on (27, 100, 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]

(127, 127+69, 121)-Net over F₂ — Digital

Digital (127, 196, 121)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(127, 127+69, 672)-Net in Base 2 — Upper bound on s

There is no (127, 196, 673)-net in base 2, because

1 times m-reduction [i] would yield (127, 195, 673)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 52353 535693 100889 433207 139964 584843 968014 914498 630228 921332 > 2¹⁹⁵ [i]