MinT - Best Known (252−95, 252, s)-Nets in Base 2

Best Known (252−95, 252, s)-Nets in Base 2

(252−95, 252, 84)-Net over F₂ — Constructive and digital

Digital (157, 252, 84)-net over F₂, using

3 times m-reduction [i] based on digital (157, 255, 84)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (54, 103, 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, 152, 42)-net over F₂, using
    - net from sequence [i] based on digital (54, 41)-sequence over F₂ (see above)

(252−95, 252, 86)-Net in Base 2 — Constructive

(157, 252, 86)-net in base 2, using

2 times m-reduction [i] based on (157, 254, 86)-net in base 2, using
- trace code for nets [i] based on (30, 127, 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]

(252−95, 252, 129)-Net over F₂ — Digital

Digital (157, 252, 129)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(252−95, 252, 677)-Net in Base 2 — Upper bound on s

There is no (157, 252, 678)-net in base 2, because

1 times m-reduction [i] would yield (157, 251, 678)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3822 752998 380483 177689 779933 970620 396077 119981 216474 395831 374676 695558 256512 > 2²⁵¹ [i]