MinT - Best Known (256−97, 256, s)-Nets in Base 2

Best Known (256−97, 256, s)-Nets in Base 2

(256−97, 256, 84)-Net over F₂ — Constructive and digital

Digital (159, 256, 84)-net over F₂, using

t-expansion [i] based on digital (158, 256, 84)-net over F₂, using
- 2 times m-reduction [i] based on digital (158, 258, 84)-net over F₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (54, 104, 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]
    2. digital (54, 154, 42)-net over F₂, using
      - net from sequence [i] based on digital (54, 41)-sequence over F₂ (see above)

(256−97, 256, 86)-Net in Base 2 — Constructive

(159, 256, 86)-net in base 2, using

2 times m-reduction [i] based on (159, 258, 86)-net in base 2, using
- trace code for nets [i] based on (30, 129, 43)-net in base 4, using
  - net from sequence [i] based on (30, 42)-sequence in base 4, using
    - base expansion [i] based on digital (60, 42)-sequence over F₂, using
      - t-expansion [i] based on digital (59, 42)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

(256−97, 256, 129)-Net over F₂ — Digital

Digital (159, 256, 129)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(256−97, 256, 676)-Net in Base 2 — Upper bound on s

There is no (159, 256, 677)-net in base 2, because

1 times m-reduction [i] would yield (159, 255, 677)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 60814 512069 389526 511926 395590 202077 413629 842242 346567 579024 015174 481068 233000 > 2²⁵⁵ [i]