MinT - Best Known (219−88, 219, s)-Nets in Base 4

Best Known (219−88, 219, s)-Nets in Base 4

(219−88, 219, 134)-Net over F₄ — Constructive and digital

Digital (131, 219, 134)-net over F₄, using

2 times m-reduction [i] based on digital (131, 221, 134)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 58, 30)-net over F₄, using
    - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
        F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (73, 163, 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]

(219−88, 219, 312)-Net over F₄ — Digital

Digital (131, 219, 312)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−88, 219, 5671)-Net in Base 4 — Upper bound on s

There is no (131, 219, 5672)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 714982 447781 544747 017454 675364 778390 102940 150998 827806 935513 164997 273963 257731 252086 777242 171206 230967 609596 252747 843070 548071 217728 > 4²¹⁹ [i]