MinT - Best Known (82−45, 82, s)-Nets in Base 4

Best Known (82−45, 82, s)-Nets in Base 4

(82−45, 82, 56)-Net over F₄ — Constructive and digital

Digital (37, 82, 56)-net over F₄, using

t-expansion [i] based on digital (33, 82, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(82−45, 82, 66)-Net over F₄ — Digital

Digital (37, 82, 66)-net over F₄, using

net from sequence [i] based on digital (37, 65)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 37 and N(F) ≥ 66, using
  - Drinfeld modules of rank 1 [i]

(82−45, 82, 479)-Net in Base 4 — Upper bound on s

There is no (37, 82, 480)-net in base 4, because

1 times m-reduction [i] would yield (37, 81, 480)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5 985449 992285 430904 436681 206933 349993 386430 528711 > 4⁸¹ [i]