MinT - Best Known (39−25, 39, s)-Nets in Base 4

Best Known (39−25, 39, s)-Nets in Base 4

(39−25, 39, 30)-Net over F₄ — Constructive and digital

Digital (14, 39, 30)-net over F₄, using

t-expansion [i] based on digital (13, 39, 30)-net over F₄, using
- net from sequence [i] based on digital (13, 29)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
    - F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(39−25, 39, 33)-Net over F₄ — Digital

Digital (14, 39, 33)-net over F₄, using

t-expansion [i] based on digital (13, 39, 33)-net over F₄, using
- net from sequence [i] based on digital (13, 32)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 33, using
    - Drinfeld modules of rank 1 [i]

(39−25, 39, 132)-Net in Base 4 — Upper bound on s

There is no (14, 39, 133)-net in base 4, because

1 times m-reduction [i] would yield (14, 38, 133)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 76224 585790 608854 721040 > 4³⁸ [i]