MinT - Best Known (111, 111+49, s)-Nets in Base 2

Best Known (111, 111+49, s)-Nets in Base 2

(111, 111+49, 77)-Net over F₂ — Constructive and digital

Digital (111, 160, 77)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (8, 32, 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]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (79, 128, 66)-net over F₂, using
  - trace code for nets [i] based on digital (15, 64, 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]

(111, 111+49, 86)-Net in Base 2 — Constructive

(111, 160, 86)-net in base 2, using

2 times m-reduction [i] based on (111, 162, 86)-net in base 2, using
- trace code for nets [i] based on (30, 81, 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]

(111, 111+49, 142)-Net over F₂ — Digital

Digital (111, 160, 142)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(111, 111+49, 932)-Net in Base 2 — Upper bound on s

There is no (111, 160, 933)-net in base 2, because

1 times m-reduction [i] would yield (111, 159, 933)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 736909 453033 785875 728332 902225 788426 755285 399892 > 2¹⁵⁹ [i]