MinT - Best Known (140−77, 140, s)-Nets in Base 4

Best Known (140−77, 140, s)-Nets in Base 4

(140−77, 140, 66)-Net over F₄ — Constructive and digital

Digital (63, 140, 66)-net over F₄, using

t-expansion [i] based on digital (49, 140, 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]

(140−77, 140, 99)-Net over F₄ — Digital

Digital (63, 140, 99)-net over F₄, using

t-expansion [i] based on digital (61, 140, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(140−77, 140, 767)-Net in Base 4 — Upper bound on s

There is no (63, 140, 768)-net in base 4, because

1 times m-reduction [i] would yield (63, 139, 768)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 504506 313576 226690 638371 293138 883380 652872 776246 703253 957211 653655 341676 260158 618105 > 4¹³⁹ [i]