MinT - Best Known (210−81, 210, s)-Nets in Base 4

Best Known (210−81, 210, s)-Nets in Base 4

(210−81, 210, 137)-Net over F₄ — Constructive and digital

Digital (129, 210, 137)-net over F₄, using

1 times m-reduction [i] based on digital (129, 211, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 56, 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, 155, 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]

(210−81, 210, 344)-Net over F₄ — Digital

Digital (129, 210, 344)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(210−81, 210, 7319)-Net in Base 4 — Upper bound on s

There is no (129, 210, 7320)-net in base 4, because

1 times m-reduction [i] would yield (129, 209, 7320)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 678049 682302 419183 422210 706012 741081 224290 987288 191808 474675 545466 635589 931217 311455 742235 996812 423390 695953 411125 324611 333720 > 4²⁰⁹ [i]