MinT - Best Known (120−77, 120, s)-Nets in Base 4

Best Known (120−77, 120, s)-Nets in Base 4

(120−77, 120, 56)-Net over F₄ — Constructive and digital

Digital (43, 120, 56)-net over F₄, using

t-expansion [i] based on digital (33, 120, 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]

(120−77, 120, 75)-Net over F₄ — Digital

Digital (43, 120, 75)-net over F₄, using

t-expansion [i] based on digital (40, 120, 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]

(120−77, 120, 354)-Net in Base 4 — Upper bound on s

There is no (43, 120, 355)-net in base 4, because

1 times m-reduction [i] would yield (43, 119, 355)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 467544 906269 878914 862006 266690 987945 526544 262415 044017 167172 565853 485616 > 4¹¹⁹ [i]