MinT - Best Known (70, 70+123, s)-Nets in Base 4

Best Known (70, 70+123, s)-Nets in Base 4

(70, 70+123, 66)-Net over F₄ — Constructive and digital

Digital (70, 193, 66)-net over F₄, using

t-expansion [i] based on digital (49, 193, 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]

(70, 70+123, 105)-Net over F₄ — Digital

Digital (70, 193, 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]

(70, 70+123, 567)-Net in Base 4 — Upper bound on s

There is no (70, 193, 568)-net in base 4, because

1 times m-reduction [i] would yield (70, 192, 568)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 39 711984 741136 524809 011865 719514 321189 619077 065473 959820 991155 917937 870475 751761 111781 144053 155388 934035 171783 147840 > 4¹⁹² [i]