MinT - Best Known (120, 120+55, s)-Nets in Base 2

Best Known (120, 120+55, s)-Nets in Base 2

(120, 120+55, 77)-Net over F₂ — Constructive and digital

Digital (120, 175, 77)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (8, 35, 11)-net over F₂, using
  - net from sequence [i] based on digital (8, 10)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 8 and N(F) ≥ 11, using
      - an explicitly constructive algebraic function field [i]
2. digital (85, 140, 66)-net over F₂, using
  - trace code for nets [i] based on digital (15, 70, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]

(120, 120+55, 86)-Net in Base 2 — Constructive

(120, 175, 86)-net in base 2, using

5 times m-reduction [i] based on (120, 180, 86)-net in base 2, using
- trace code for nets [i] based on (30, 90, 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]

(120, 120+55, 143)-Net over F₂ — Digital

Digital (120, 175, 143)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(120, 120+55, 912)-Net in Base 2 — Upper bound on s

There is no (120, 175, 913)-net in base 2, because

1 times m-reduction [i] would yield (120, 174, 913)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 24506 869251 752622 188201 163702 580758 363571 440463 954304 > 2¹⁷⁴ [i]