MinT - Best Known (249−105, 249, s)-Nets in Base 4

Best Known (249−105, 249, s)-Nets in Base 4

(249−105, 249, 137)-Net over F₄ — Constructive and digital

Digital (144, 249, 137)-net over F₄, using

7 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]

(249−105, 249, 299)-Net over F₄ — Digital

Digital (144, 249, 299)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(249−105, 249, 4970)-Net in Base 4 — Upper bound on s

There is no (144, 249, 4971)-net in base 4, because

1 times m-reduction [i] would yield (144, 248, 4971)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 205051 797854 302976 647748 061544 108205 955841 978298 305813 210370 660882 797908 635862 011881 811642 482855 648972 155059 174638 742021 498840 071274 049373 022746 525240 > 4²⁴⁸ [i]