MinT - Best Known (228−129, 228, s)-Nets in Base 4

Best Known (228−129, 228, s)-Nets in Base 4

(228−129, 228, 104)-Net over F₄ — Constructive and digital

Digital (99, 228, 104)-net over F₄, using

t-expansion [i] based on digital (73, 228, 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]

(228−129, 228, 144)-Net over F₄ — Digital

Digital (99, 228, 144)-net over F₄, using

t-expansion [i] based on digital (91, 228, 144)-net over F₄, using
- net from sequence [i] based on digital (91, 143)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 91 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(228−129, 228, 1071)-Net in Base 4 — Upper bound on s

There is no (99, 228, 1072)-net in base 4, because

1 times m-reduction [i] would yield (99, 227, 1072)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 47764 390027 730873 614577 667551 283150 496277 802332 050437 848514 648207 132592 235928 638225 676254 253129 180305 769045 067983 072964 912987 577865 955792 > 4²²⁷ [i]