MinT - Best Known (199−139, 199, s)-Nets in Base 4

Best Known (199−139, 199, s)-Nets in Base 4

(199−139, 199, 66)-Net over F₄ — Constructive and digital

Digital (60, 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]

(199−139, 199, 91)-Net over F₄ — Digital

Digital (60, 199, 91)-net over F₄, using

t-expansion [i] based on digital (50, 199, 91)-net over F₄, using
- net from sequence [i] based on digital (50, 90)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 50 and N(F) ≥ 91, using
    - a curve from the manYPoints database [i]

(199−139, 199, 417)-Net in Base 4 — Upper bound on s

There is no (60, 199, 418)-net in base 4, because

1 times m-reduction [i] would yield (60, 198, 418)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 167365 344251 276144 194827 414832 666835 014799 751202 437622 110398 370687 532036 442780 342096 214957 388013 482636 588491 808487 755600 > 4¹⁹⁸ [i]