MinT - Best Known (172−101, 172, s)-Nets in Base 4

Best Known (172−101, 172, s)-Nets in Base 4

(172−101, 172, 66)-Net over F₄ — Constructive and digital

Digital (71, 172, 66)-net over F₄, using

t-expansion [i] based on digital (49, 172, 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]

(172−101, 172, 105)-Net over F₄ — Digital

Digital (71, 172, 105)-net over F₄, using

t-expansion [i] based on digital (70, 172, 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]

(172−101, 172, 703)-Net in Base 4 — Upper bound on s

There is no (71, 172, 704)-net in base 4, because

1 times m-reduction [i] would yield (71, 171, 704)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9 140674 569192 759869 093100 733277 519577 460210 564349 947696 622449 636355 664217 932117 664759 949700 501904 681315 > 4¹⁷¹ [i]