MinT - Best Known (251−171, 251, s)-Nets in Base 4

Best Known (251−171, 251, s)-Nets in Base 4

(251−171, 251, 104)-Net over F₄ — Constructive and digital

Digital (80, 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]

(251−171, 251, 112)-Net over F₄ — Digital

Digital (80, 251, 112)-net over F₄, using

t-expansion [i] based on digital (73, 251, 112)-net over F₄, using
- net from sequence [i] based on digital (73, 111)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 112, using
    - Drinfeld modules of rank 1 [i]

(251−171, 251, 570)-Net in Base 4 — Upper bound on s

There is no (80, 251, 571)-net in base 4, because

1 times m-reduction [i] would yield (80, 250, 571)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 519342 720808 957596 762469 815887 273542 648152 594346 491721 885958 744233 797618 251509 004898 917584 565858 947295 811457 195811 656947 925029 829021 660253 057054 284000 > 4²⁵⁰ [i]