MinT - Best Known (115, 166, s)-Nets in Base 2

Best Known (115, 166, s)-Nets in Base 2

(115, 166, 78)-Net over F₂ — Constructive and digital

Digital (115, 166, 78)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (9, 34, 12)-net over F₂, using
  - net from sequence [i] based on digital (9, 11)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 9 and N(F) ≥ 12, using
      - an explicitly constructive algebraic function field [i]
2. digital (81, 132, 66)-net over F₂, using
  - trace code for nets [i] based on digital (15, 66, 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]

(115, 166, 86)-Net in Base 2 — Constructive

(115, 166, 86)-net in base 2, using

4 times m-reduction [i] based on (115, 170, 86)-net in base 2, using
- trace code for nets [i] based on (30, 85, 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]

(115, 166, 145)-Net over F₂ — Digital

Digital (115, 166, 145)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(115, 166, 950)-Net in Base 2 — Upper bound on s

There is no (115, 166, 951)-net in base 2, because

1 times m-reduction [i] would yield (115, 165, 951)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 46 930870 463101 168525 143147 163557 420299 015225 109920 > 2¹⁶⁵ [i]