MinT - Best Known (109−65, 109, s)-Nets in Base 4

Best Known (109−65, 109, s)-Nets in Base 4

(109−65, 109, 56)-Net over F₄ — Constructive and digital

Digital (44, 109, 56)-net over F₄, using

t-expansion [i] based on digital (33, 109, 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]

(109−65, 109, 75)-Net over F₄ — Digital

Digital (44, 109, 75)-net over F₄, using

t-expansion [i] based on digital (40, 109, 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]

(109−65, 109, 433)-Net in Base 4 — Upper bound on s

There is no (44, 109, 434)-net in base 4, because

1 times m-reduction [i] would yield (44, 108, 434)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 110712 290339 267665 211907 664296 896024 192938 049573 762708 737065 643675 > 4¹⁰⁸ [i]