MinT - Best Known (227−115, 227, s)-Nets in Base 4

Best Known (227−115, 227, s)-Nets in Base 4

(227−115, 227, 130)-Net over F₄ — Constructive and digital

Digital (112, 227, 130)-net over F₄, using

t-expansion [i] based on digital (105, 227, 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]

(227−115, 227, 165)-Net over F₄ — Digital

Digital (112, 227, 165)-net over F₄, using

t-expansion [i] based on digital (109, 227, 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]

(227−115, 227, 1748)-Net in Base 4 — Upper bound on s

There is no (112, 227, 1749)-net in base 4, because

1 times m-reduction [i] would yield (112, 226, 1749)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11946 327468 362769 607795 156379 094863 397350 995767 185872 748488 724580 135138 215066 620811 693336 737698 471433 365926 304483 417528 090442 608295 890880 > 4²²⁶ [i]