MinT - Best Known (199−133, 199, s)-Nets in Base 4

Best Known (199−133, 199, s)-Nets in Base 4

(199−133, 199, 66)-Net over F₄ — Constructive and digital

Digital (66, 199, 66)-net over F₄, using

t-expansion [i] based on digital (49, 199, 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]

(199−133, 199, 99)-Net over F₄ — Digital

Digital (66, 199, 99)-net over F₄, using

t-expansion [i] based on digital (61, 199, 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]

(199−133, 199, 489)-Net in Base 4 — Upper bound on s

There is no (66, 199, 490)-net in base 4, because

1 times m-reduction [i] would yield (66, 198, 490)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 167965 001527 353374 411504 960209 732206 810242 264014 806947 054034 323353 390841 470024 017432 773291 827103 864192 920010 922402 558700 > 4¹⁹⁸ [i]