MinT - Best Known (83−45, 83, s)-Nets in Base 4

Best Known (83−45, 83, s)-Nets in Base 4

(83−45, 83, 56)-Net over F₄ — Constructive and digital

Digital (38, 83, 56)-net over F₄, using

t-expansion [i] based on digital (33, 83, 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]

(83−45, 83, 66)-Net over F₄ — Digital

Digital (38, 83, 66)-net over F₄, using

t-expansion [i] based on digital (37, 83, 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]

(83−45, 83, 511)-Net in Base 4 — Upper bound on s

There is no (38, 83, 512)-net in base 4, because

1 times m-reduction [i] would yield (38, 82, 512)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 23 596867 195214 128717 588913 647390 049912 847633 478945 > 4⁸² [i]