MinT - Best Known (211−85, 211, s)-Nets in Base 4

Best Known (211−85, 211, s)-Nets in Base 4

(211−85, 211, 131)-Net over F₄ — Constructive and digital

Digital (126, 211, 131)-net over F₄, using

1 times m-reduction [i] based on digital (126, 212, 131)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 53, 27)-net over F₄, using
    - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
        an explicitly constructive algebraic function field [i]
  2. digital (73, 159, 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]

(211−85, 211, 299)-Net over F₄ — Digital

Digital (126, 211, 299)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(211−85, 211, 5601)-Net in Base 4 — Upper bound on s

There is no (126, 211, 5602)-net in base 4, because

1 times m-reduction [i] would yield (126, 210, 5602)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 713644 403594 899060 952791 833910 985956 225497 555205 698509 680470 108702 388131 707109 452112 621988 199843 140221 039532 001633 118333 949280 > 4²¹⁰ [i]