MinT - Best Known (195−71, 195, s)-Nets in Base 2

Best Known (195−71, 195, s)-Nets in Base 2

(195−71, 195, 69)-Net over F₂ — Constructive and digital

Digital (124, 195, 69)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (19, 54, 20)-net over F₂, using
  - net from sequence [i] based on digital (19, 19)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 19 and N(F) ≥ 20, using
      - an explicitly constructive algebraic function field [i]
2. digital (70, 141, 49)-net over F₂, using
  - net from sequence [i] based on digital (70, 48)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, and 1 place with degree 2 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(195−71, 195, 72)-Net in Base 2 — Constructive

(124, 195, 72)-net in base 2, using

1 times m-reduction [i] based on (124, 196, 72)-net in base 2, using
- trace code for nets [i] based on (26, 98, 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]

(195−71, 195, 112)-Net over F₂ — Digital

Digital (124, 195, 112)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(195−71, 195, 598)-Net in Base 2 — Upper bound on s

There is no (124, 195, 599)-net in base 2, because

1 times m-reduction [i] would yield (124, 194, 599)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 26207 188967 554199 990825 963488 003018 021589 534073 774669 292380 > 2¹⁹⁴ [i]