MinT - Best Known (216−137, 216, s)-Nets in Base 4

Best Known (216−137, 216, s)-Nets in Base 4

(216−137, 216, 104)-Net over F₄ — Constructive and digital

Digital (79, 216, 104)-net over F₄, using

t-expansion [i] based on digital (73, 216, 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]

(216−137, 216, 112)-Net over F₄ — Digital

Digital (79, 216, 112)-net over F₄, using

t-expansion [i] based on digital (73, 216, 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]

(216−137, 216, 643)-Net in Base 4 — Upper bound on s

There is no (79, 216, 644)-net in base 4, because

1 times m-reduction [i] would yield (79, 215, 644)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2841 946814 884897 504412 671049 042244 616493 180723 370496 408909 961024 710453 700444 468593 765007 396533 254722 811723 526304 054658 525819 275584 > 4²¹⁵ [i]