MinT - Best Known (243−102, 243, s)-Nets in Base 4

Best Known (243−102, 243, s)-Nets in Base 4

(243−102, 243, 137)-Net over F₄ — Constructive and digital

Digital (141, 243, 137)-net over F₄, using

4 times m-reduction [i] based on digital (141, 247, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 68, 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, 179, 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]

(243−102, 243, 297)-Net over F₄ — Digital

Digital (141, 243, 297)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(243−102, 243, 4848)-Net in Base 4 — Upper bound on s

There is no (141, 243, 4849)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 200 077218 326655 309131 358418 180796 027283 610349 638481 508377 272189 913047 816125 167410 581640 427560 410965 331427 711790 739151 550951 923760 482176 103819 941760 > 4²⁴³ [i]