MinT - Best Known (72, 72+109, s)-Nets in Base 4

Best Known (72, 72+109, s)-Nets in Base 4

(72, 72+109, 66)-Net over F₄ — Constructive and digital

Digital (72, 181, 66)-net over F₄, using

t-expansion [i] based on digital (49, 181, 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+109, 105)-Net over F₄ — Digital

Digital (72, 181, 105)-net over F₄, using

t-expansion [i] based on digital (70, 181, 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+109, 666)-Net in Base 4 — Upper bound on s

There is no (72, 181, 667)-net in base 4, because

1 times m-reduction [i] would yield (72, 180, 667)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 424604 052042 013936 045691 563529 339032 690134 228596 085798 361432 661872 390224 576510 727981 534462 435929 788717 935160 > 4¹⁸⁰ [i]