MinT - Best Known (117−81, 117, s)-Nets in Base 4

Best Known (117−81, 117, s)-Nets in Base 4

(117−81, 117, 56)-Net over F₄ — Constructive and digital

Digital (36, 117, 56)-net over F₄, using

t-expansion [i] based on digital (33, 117, 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]

(117−81, 117, 65)-Net over F₄ — Digital

Digital (36, 117, 65)-net over F₄, using

t-expansion [i] based on digital (33, 117, 65)-net over F₄, using
- net from sequence [i] based on digital (33, 64)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 65, using
    - Drinfeld modules of rank 1 [i]

(117−81, 117, 261)-Net in Base 4 — Upper bound on s

There is no (36, 117, 262)-net in base 4, because

1 times m-reduction [i] would yield (36, 116, 262)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7544 909907 710136 094939 098011 100584 344644 822692 978376 054603 823578 222880 > 4¹¹⁶ [i]