MinT - Best Known (233−157, 233, s)-Nets in Base 4

Best Known (233−157, 233, s)-Nets in Base 4

(233−157, 233, 104)-Net over F₄ — Constructive and digital

Digital (76, 233, 104)-net over F₄, using

t-expansion [i] based on digital (73, 233, 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]

(233−157, 233, 112)-Net over F₄ — Digital

Digital (76, 233, 112)-net over F₄, using

t-expansion [i] based on digital (73, 233, 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]

(233−157, 233, 552)-Net in Base 4 — Upper bound on s

There is no (76, 233, 553)-net in base 4, because

1 times m-reduction [i] would yield (76, 232, 553)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 49 732778 173663 079261 973226 307588 873969 087374 542791 586372 152385 508589 536728 637285 443675 404460 311158 415161 051714 709558 059128 395881 839902 032000 > 4²³² [i]