MinT - Best Known (128, 128+69, s)-Nets in Base 2

Best Known (128, 128+69, s)-Nets in Base 2

(128, 128+69, 75)-Net over F₂ — Constructive and digital

Digital (128, 197, 75)-net over F₂, using

1 times m-reduction [i] based on digital (128, 198, 75)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 74, 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, 124, 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]

(128, 128+69, 84)-Net in Base 2 — Constructive

(128, 197, 84)-net in base 2, using

5 times m-reduction [i] based on (128, 202, 84)-net in base 2, using
- trace code for nets [i] based on (27, 101, 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]

(128, 128+69, 123)-Net over F₂ — Digital

Digital (128, 197, 123)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(128, 128+69, 686)-Net in Base 2 — Upper bound on s

There is no (128, 197, 687)-net in base 2, because

1 times m-reduction [i] would yield (128, 196, 687)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 100721 355694 025086 369217 640462 356836 672758 447790 808657 442423 > 2¹⁹⁶ [i]