MinT - Best Known (109−62, 109, s)-Nets in Base 4

Best Known (109−62, 109, s)-Nets in Base 4

(109−62, 109, 56)-Net over F₄ — Constructive and digital

Digital (47, 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]

(109−62, 109, 81)-Net over F₄ — Digital

Digital (47, 109, 81)-net over F₄, using

t-expansion [i] based on digital (46, 109, 81)-net over F₄, using
- net from sequence [i] based on digital (46, 80)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 46 and N(F) ≥ 81, using
    - Drinfeld modules of rank 1 [i]

(109−62, 109, 516)-Net in Base 4 — Upper bound on s

There is no (47, 109, 517)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 423601 540401 695037 460633 465456 437406 348102 111489 597375 290840 630784 > 4¹⁰⁹ [i]