MinT - Best Known (228−89, 228, s)-Nets in Base 4

Best Known (228−89, 228, s)-Nets in Base 4

(228−89, 228, 138)-Net over F₄ — Constructive and digital

Digital (139, 228, 138)-net over F₄, using

1 times m-reduction [i] based on digital (139, 229, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 66, 34)-net over F₄, using
    - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
        T₅ from the second 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]

(228−89, 228, 356)-Net over F₄ — Digital

Digital (139, 228, 356)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(228−89, 228, 7307)-Net in Base 4 — Upper bound on s

There is no (139, 228, 7308)-net in base 4, because

1 times m-reduction [i] would yield (139, 227, 7308)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 46776 730929 351795 305329 376004 836136 133615 025673 408544 745615 957831 417254 242987 557870 821752 631353 459539 911163 012339 406884 141416 848699 206560 > 4²²⁷ [i]