MinT - Best Known (70, 70+97, s)-Nets in Base 4

Best Known (70, 70+97, s)-Nets in Base 4

(70, 70+97, 66)-Net over F₄ — Constructive and digital

Digital (70, 167, 66)-net over F₄, using

t-expansion [i] based on digital (49, 167, 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+97, 105)-Net over F₄ — Digital

Digital (70, 167, 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+97, 716)-Net in Base 4 — Upper bound on s

There is no (70, 167, 717)-net in base 4, because

1 times m-reduction [i] would yield (70, 166, 717)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9322 650321 528006 013152 535930 867331 030191 558430 636081 708933 111110 463405 913561 277091 423887 100080 437196 > 4¹⁶⁶ [i]