MinT - Best Known (86−45, 86, s)-Nets in Base 4

Best Known (86−45, 86, s)-Nets in Base 4

(86−45, 86, 56)-Net over F₄ — Constructive and digital

Digital (41, 86, 56)-net over F₄, using

t-expansion [i] based on digital (33, 86, 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]

(86−45, 86, 75)-Net over F₄ — Digital

Digital (41, 86, 75)-net over F₄, using

t-expansion [i] based on digital (40, 86, 75)-net over F₄, using
- net from sequence [i] based on digital (40, 74)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 40 and N(F) ≥ 75, using
    - Drinfeld modules of rank 1 [i]

(86−45, 86, 621)-Net in Base 4 — Upper bound on s

There is no (41, 86, 622)-net in base 4, because

1 times m-reduction [i] would yield (41, 85, 622)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1501 569810 107094 056247 025644 518677 892307 100088 491960 > 4⁸⁵ [i]