MinT - Best Known (251−173, 251, s)-Nets in Base 4

Best Known (251−173, 251, s)-Nets in Base 4

(251−173, 251, 104)-Net over F₄ — Constructive and digital

Digital (78, 251, 104)-net over F₄, using

t-expansion [i] based on digital (73, 251, 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]

(251−173, 251, 112)-Net over F₄ — Digital

Digital (78, 251, 112)-net over F₄, using

t-expansion [i] based on digital (73, 251, 112)-net over F₄, using
- net from sequence [i] based on digital (73, 111)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 112, using
    - Drinfeld modules of rank 1 [i]

(251−173, 251, 547)-Net in Base 4 — Upper bound on s

There is no (78, 251, 548)-net in base 4, because

1 times m-reduction [i] would yield (78, 250, 548)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 671251 569552 481898 496059 002223 548816 211272 385055 459904 859832 337507 437809 138397 005546 043136 349890 259547 393928 025313 700875 209889 792153 965347 939231 234240 > 4²⁵⁰ [i]