MinT - Best Known (199−73, 199, s)-Nets in Base 2

Best Known (199−73, 199, s)-Nets in Base 2

(199−73, 199, 69)-Net over F₂ — Constructive and digital

Digital (126, 199, 69)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (39, 75, 33)-net over F₂, using
  - net from sequence [i] based on digital (39, 32)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 39 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (51, 124, 36)-net over F₂, using
  - net from sequence [i] based on digital (51, 35)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 3 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

(199−73, 199, 72)-Net in Base 2 — Constructive

(126, 199, 72)-net in base 2, using

1 times m-reduction [i] based on (126, 200, 72)-net in base 2, using
- trace code for nets [i] based on (26, 100, 36)-net in base 4, using
  - net from sequence [i] based on (26, 35)-sequence in base 4, using
    - base expansion [i] based on digital (52, 35)-sequence over F₂, using
      - t-expansion [i] based on digital (51, 35)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 3 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

(199−73, 199, 112)-Net over F₂ — Digital

Digital (126, 199, 112)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(199−73, 199, 594)-Net in Base 2 — Upper bound on s

There is no (126, 199, 595)-net in base 2, because

1 times m-reduction [i] would yield (126, 198, 595)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 408734 016861 879151 343324 990485 946946 620759 418915 919338 580680 > 2¹⁹⁸ [i]