MinT - Best Known (251−159, 251, s)-Nets in Base 4

Best Known (251−159, 251, s)-Nets in Base 4

(251−159, 251, 104)-Net over F₄ — Constructive and digital

Digital (92, 251, 104)-net over F₄, using

t-expansion [i] based on digital (73, 251, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(251−159, 251, 144)-Net over F₄ — Digital

Digital (92, 251, 144)-net over F₄, using

t-expansion [i] based on digital (91, 251, 144)-net over F₄, using
- net from sequence [i] based on digital (91, 143)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 91 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(251−159, 251, 746)-Net in Base 4 — Upper bound on s

There is no (92, 251, 747)-net in base 4, because

1 times m-reduction [i] would yield (92, 250, 747)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 376244 681030 704925 550670 640671 292349 487121 101403 979299 663440 756677 213419 289716 994545 135565 935725 123791 051218 719118 810097 861456 192582 649759 381019 967424 > 4²⁵⁰ [i]