MinT - Best Known (189−117, 189, s)-Nets in Base 4

Best Known (189−117, 189, s)-Nets in Base 4

(189−117, 189, 66)-Net over F₄ — Constructive and digital

Digital (72, 189, 66)-net over F₄, using

t-expansion [i] based on digital (49, 189, 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]

(189−117, 189, 105)-Net over F₄ — Digital

Digital (72, 189, 105)-net over F₄, using

t-expansion [i] based on digital (70, 189, 105)-net over F₄, using
- net from sequence [i] based on digital (70, 104)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 70 and N(F) ≥ 105, using
    - Drinfeld modules of rank 1 [i]

(189−117, 189, 622)-Net in Base 4 — Upper bound on s

There is no (72, 189, 623)-net in base 4, because

1 times m-reduction [i] would yield (72, 188, 623)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 157833 976816 816504 797202 055757 940563 316424 079844 098652 956352 444982 849095 292327 490965 432321 755770 191275 900515 241780 > 4¹⁸⁸ [i]