MinT - Best Known (191−39, 191, s)-Nets in Base 4

Best Known (191−39, 191, s)-Nets in Base 4

(191−39, 191, 1061)-Net over F₄ — Constructive and digital

Digital (152, 191, 1061)-net over F₄, using

4¹ times duplication [i] based on digital (151, 190, 1061)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 34, 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]
  2. digital (117, 156, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 39, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(191−39, 191, 5338)-Net over F₄ — Digital

Digital (152, 191, 5338)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(191−39, 191, 2771352)-Net in Base 4 — Upper bound on s

There is no (152, 191, 2771353)-net in base 4, because

1 times m-reduction [i] would yield (152, 190, 2771353)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 462634 578007 160341 536718 953585 628121 203019 987368 786859 867196 591587 707289 182734 006532 589294 006541 389136 751862 330504 > 4¹⁹⁰ [i]