MinT - Best Known (81−45, 81, s)-Nets in Base 4

Best Known (81−45, 81, s)-Nets in Base 4

(81−45, 81, 56)-Net over F₄ — Constructive and digital

Digital (36, 81, 56)-net over F₄, using

t-expansion [i] based on digital (33, 81, 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]

(81−45, 81, 65)-Net over F₄ — Digital

Digital (36, 81, 65)-net over F₄, using

t-expansion [i] based on digital (33, 81, 65)-net over F₄, using
- net from sequence [i] based on digital (33, 64)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 65, using
    - Drinfeld modules of rank 1 [i]

(81−45, 81, 449)-Net in Base 4 — Upper bound on s

There is no (36, 81, 450)-net in base 4, because

1 times m-reduction [i] would yield (36, 80, 450)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 522806 782095 199196 156786 419503 414861 623870 024736 > 4⁸⁰ [i]