MinT - Best Known (144, 144+111, s)-Nets in Base 4

Best Known (144, 144+111, s)-Nets in Base 4

(144, 144+111, 137)-Net over F₄ — Constructive and digital

Digital (144, 255, 137)-net over F₄, using

1 times m-reduction [i] based on digital (144, 256, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 71, 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, 185, 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]

(144, 144+111, 276)-Net over F₄ — Digital

Digital (144, 255, 276)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(144, 144+111, 4244)-Net in Base 4 — Upper bound on s

There is no (144, 255, 4245)-net in base 4, because

1 times m-reduction [i] would yield (144, 254, 4245)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 839 423338 965771 640239 687756 694109 182120 008021 470421 251416 460148 933971 358636 536482 583714 178768 797634 980788 027677 914994 719079 733300 439920 835069 812004 449408 > 4²⁵⁴ [i]