MinT - Best Known (190−127, 190, s)-Nets in Base 4

Best Known (190−127, 190, s)-Nets in Base 4

(190−127, 190, 66)-Net over F₄ — Constructive and digital

Digital (63, 190, 66)-net over F₄, using

t-expansion [i] based on digital (49, 190, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(190−127, 190, 99)-Net over F₄ — Digital

Digital (63, 190, 99)-net over F₄, using

t-expansion [i] based on digital (61, 190, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(190−127, 190, 468)-Net in Base 4 — Upper bound on s

There is no (63, 190, 469)-net in base 4, because

1 times m-reduction [i] would yield (63, 189, 469)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 664806 502664 003252 138211 336737 527289 240344 012415 585373 582942 793671 986290 579233 846588 992899 179931 387553 123916 871360 > 4¹⁸⁹ [i]