MinT - Best Known (231−121, 231, s)-Nets in Base 4

Best Known (231−121, 231, s)-Nets in Base 4

(231−121, 231, 130)-Net over F₄ — Constructive and digital

Digital (110, 231, 130)-net over F₄, using

t-expansion [i] based on digital (105, 231, 130)-net over F₄, using
- net from sequence [i] based on digital (105, 129)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 105 and N(F) ≥ 130, using
    - T₇ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(231−121, 231, 165)-Net over F₄ — Digital

Digital (110, 231, 165)-net over F₄, using

t-expansion [i] based on digital (109, 231, 165)-net over F₄, using
- net from sequence [i] based on digital (109, 164)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 109 and N(F) ≥ 165, using
    - Drinfeld modules of rank 1 [i]

(231−121, 231, 1521)-Net in Base 4 — Upper bound on s

There is no (110, 231, 1522)-net in base 4, because

1 times m-reduction [i] would yield (110, 230, 1522)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 993923 731711 332123 450725 395500 736629 658707 834717 007916 305079 185338 984596 462899 485265 259788 126850 278006 243799 912009 936440 790874 414011 536496 > 4²³⁰ [i]