MinT - Best Known (211−137, 211, s)-Nets in Base 4

Best Known (211−137, 211, s)-Nets in Base 4

(211−137, 211, 104)-Net over F₄ — Constructive and digital

Digital (74, 211, 104)-net over F₄, using

t-expansion [i] based on digital (73, 211, 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]

(211−137, 211, 112)-Net over F₄ — Digital

Digital (74, 211, 112)-net over F₄, using

t-expansion [i] based on digital (73, 211, 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]

(211−137, 211, 576)-Net in Base 4 — Upper bound on s

There is no (74, 211, 577)-net in base 4, because

1 times m-reduction [i] would yield (74, 210, 577)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 936761 850172 860373 518214 285262 922590 634558 503514 721712 884984 113761 216474 657004 864813 850659 850441 686355 725005 629879 414915 034240 > 4²¹⁰ [i]