MinT - Best Known (219−84, 219, s)-Nets in Base 4

Best Known (219−84, 219, s)-Nets in Base 4

(219−84, 219, 137)-Net over F₄ — Constructive and digital

Digital (135, 219, 137)-net over F₄, using

10 times m-reduction [i] based on digital (135, 229, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 62, 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, 167, 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]

(219−84, 219, 364)-Net over F₄ — Digital

Digital (135, 219, 364)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−84, 219, 7551)-Net in Base 4 — Upper bound on s

There is no (135, 219, 7552)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 712872 218620 235289 170615 741204 961413 637651 953406 021330 039372 482151 694080 753891 608417 066663 087752 631063 974744 076416 917569 943497 944525 > 4²¹⁹ [i]