MinT - Best Known (109−72, 109, s)-Nets in Base 4

Best Known (109−72, 109, s)-Nets in Base 4

(109−72, 109, 56)-Net over F₄ — Constructive and digital

Digital (37, 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]

(109−72, 109, 66)-Net over F₄ — Digital

Digital (37, 109, 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]

(109−72, 109, 288)-Net in Base 4 — Upper bound on s

There is no (37, 109, 289)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 466045 711721 569424 365522 450570 395769 693471 560124 935276 801046 691008 > 4¹⁰⁹ [i]