MinT - Best Known (72, 72+101, s)-Nets in Base 4

Best Known (72, 72+101, s)-Nets in Base 4

(72, 72+101, 66)-Net over F₄ — Constructive and digital

Digital (72, 173, 66)-net over F₄, using

t-expansion [i] based on digital (49, 173, 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]

(72, 72+101, 105)-Net over F₄ — Digital

Digital (72, 173, 105)-net over F₄, using

t-expansion [i] based on digital (70, 173, 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]

(72, 72+101, 724)-Net in Base 4 — Upper bound on s

There is no (72, 173, 725)-net in base 4, because

1 times m-reduction [i] would yield (72, 172, 725)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 36 794976 186304 844316 257454 846311 493503 755072 521177 482599 380400 736692 662203 446802 050994 570612 871809 467060 > 4¹⁷² [i]