MinT - Best Known (112−69, 112, s)-Nets in Base 4

Best Known (112−69, 112, s)-Nets in Base 4

(112−69, 112, 56)-Net over F₄ — Constructive and digital

Digital (43, 112, 56)-net over F₄, using

t-expansion [i] based on digital (33, 112, 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]

(112−69, 112, 75)-Net over F₄ — Digital

Digital (43, 112, 75)-net over F₄, using

t-expansion [i] based on digital (40, 112, 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]

(112−69, 112, 389)-Net in Base 4 — Upper bound on s

There is no (43, 112, 390)-net in base 4, because

1 times m-reduction [i] would yield (43, 111, 390)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 6 945286 728734 381378 973648 106343 396184 917551 356079 902857 238002 302120 > 4¹¹¹ [i]