MinT - Best Known (126−81, 126, s)-Nets in Base 4

Best Known (126−81, 126, s)-Nets in Base 4

(126−81, 126, 56)-Net over F₄ — Constructive and digital

Digital (45, 126, 56)-net over F₄, using

t-expansion [i] based on digital (33, 126, 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]

(126−81, 126, 80)-Net over F₄ — Digital

Digital (45, 126, 80)-net over F₄, using

net from sequence [i] based on digital (45, 79)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 45 and N(F) ≥ 80, using
  - Drinfeld modules of rank 1 [i]

(126−81, 126, 368)-Net in Base 4 — Upper bound on s

There is no (45, 126, 369)-net in base 4, because

1 times m-reduction [i] would yield (45, 125, 369)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1959 952311 645739 048359 600970 933019 913606 454909 101110 585208 249816 579824 568502 > 4¹²⁵ [i]