MinT - Best Known (78−39, 78, s)-Nets in Base 4

Best Known (78−39, 78, s)-Nets in Base 4

(78−39, 78, 56)-Net over F₄ — Constructive and digital

Digital (39, 78, 56)-net over F₄, using

t-expansion [i] based on digital (33, 78, 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]

(78−39, 78, 66)-Net over F₄ — Digital

Digital (39, 78, 66)-net over F₄, using

t-expansion [i] based on digital (37, 78, 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]

(78−39, 78, 712)-Net in Base 4 — Upper bound on s

There is no (39, 78, 713)-net in base 4, because

1 times m-reduction [i] would yield (39, 77, 713)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 23071 823676 832910 005849 612102 518361 551929 548000 > 4⁷⁷ [i]