MinT - Best Known (227−135, 227, s)-Nets in Base 4

Best Known (227−135, 227, s)-Nets in Base 4

(227−135, 227, 104)-Net over F₄ — Constructive and digital

Digital (92, 227, 104)-net over F₄, using

t-expansion [i] based on digital (73, 227, 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]

(227−135, 227, 144)-Net over F₄ — Digital

Digital (92, 227, 144)-net over F₄, using

t-expansion [i] based on digital (91, 227, 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]

(227−135, 227, 868)-Net in Base 4 — Upper bound on s

There is no (92, 227, 869)-net in base 4, because

1 times m-reduction [i] would yield (92, 226, 869)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11940 384487 417848 241521 465517 514502 523086 309970 362958 374171 793208 470443 795837 503047 669535 089350 553193 209361 781463 848800 985857 370780 394780 > 4²²⁶ [i]