MinT - Best Known (42, 42+69, s)-Nets in Base 4

Best Known (42, 42+69, s)-Nets in Base 4

(42, 42+69, 56)-Net over F₄ — Constructive and digital

Digital (42, 111, 56)-net over F₄, using

t-expansion [i] based on digital (33, 111, 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]

(42, 42+69, 75)-Net over F₄ — Digital

Digital (42, 111, 75)-net over F₄, using

t-expansion [i] based on digital (40, 111, 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]

(42, 42+69, 373)-Net in Base 4 — Upper bound on s

There is no (42, 111, 374)-net in base 4, because

1 times m-reduction [i] would yield (42, 110, 374)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 833148 429703 224102 823197 403901 579318 765587 777561 004974 164888 365754 > 4¹¹⁰ [i]