MinT - Best Known (237−137, 237, s)-Nets in Base 4

Best Known (237−137, 237, s)-Nets in Base 4

(237−137, 237, 104)-Net over F₄ — Constructive and digital

Digital (100, 237, 104)-net over F₄, using

t-expansion [i] based on digital (73, 237, 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]

(237−137, 237, 144)-Net over F₄ — Digital

Digital (100, 237, 144)-net over F₄, using

t-expansion [i] based on digital (91, 237, 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]

(237−137, 237, 1016)-Net in Base 4 — Upper bound on s

There is no (100, 237, 1017)-net in base 4, because

1 times m-reduction [i] would yield (100, 236, 1017)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12736 608176 829704 885531 039375 656331 069983 250992 600657 775477 887722 081389 651480 002307 125418 146033 738422 696134 407464 618862 311643 803510 311540 872666 > 4²³⁶ [i]