MinT - Best Known (212−81, 212, s)-Nets in Base 4

Best Known (212−81, 212, s)-Nets in Base 4

(212−81, 212, 137)-Net over F₄ — Constructive and digital

Digital (131, 212, 137)-net over F₄, using

5 times m-reduction [i] based on digital (131, 217, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 58, 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 (73, 159, 104)-net over F₄, using
    - net from sequence [i] based on digital (73, 103)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
        F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(212−81, 212, 359)-Net over F₄ — Digital

Digital (131, 212, 359)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−81, 212, 7847)-Net in Base 4 — Upper bound on s

There is no (131, 212, 7848)-net in base 4, because

1 times m-reduction [i] would yield (131, 211, 7848)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10 863643 316039 667761 390055 411005 213565 966467 728476 960419 962205 102877 834323 694827 618217 288453 806936 856240 811406 802706 323665 287408 > 4²¹¹ [i]