MinT - Best Known (72, 72+99, s)-Nets in Base 4

Best Known (72, 72+99, s)-Nets in Base 4

(72, 72+99, 66)-Net over F₄ — Constructive and digital

Digital (72, 171, 66)-net over F₄, using

t-expansion [i] based on digital (49, 171, 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+99, 105)-Net over F₄ — Digital

Digital (72, 171, 105)-net over F₄, using

t-expansion [i] based on digital (70, 171, 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+99, 742)-Net in Base 4 — Upper bound on s

There is no (72, 171, 743)-net in base 4, because

1 times m-reduction [i] would yield (72, 170, 743)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 383258 874520 720669 693591 451660 752555 285529 798005 746972 611049 388527 312184 926991 407495 856457 497772 827064 > 4¹⁷⁰ [i]