MinT - Best Known (114−69, 114, s)-Nets in Base 4

Best Known (114−69, 114, s)-Nets in Base 4

(114−69, 114, 56)-Net over F₄ — Constructive and digital

Digital (45, 114, 56)-net over F₄, using

t-expansion [i] based on digital (33, 114, 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]

(114−69, 114, 80)-Net over F₄ — Digital

Digital (45, 114, 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]

(114−69, 114, 424)-Net in Base 4 — Upper bound on s

There is no (45, 114, 425)-net in base 4, because

1 times m-reduction [i] would yield (45, 113, 425)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 107 911349 292092 550451 582426 494016 547356 660355 345784 977867 766105 620812 > 4¹¹³ [i]