MinT - Best Known (70, 70+103, s)-Nets in Base 4

Best Known (70, 70+103, s)-Nets in Base 4

(70, 70+103, 66)-Net over F₄ — Constructive and digital

Digital (70, 173, 66)-net over F₄, using

t-expansion [i] based on digital (49, 173, 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+103, 105)-Net over F₄ — Digital

Digital (70, 173, 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+103, 668)-Net in Base 4 — Upper bound on s

There is no (70, 173, 669)-net in base 4, because

1 times m-reduction [i] would yield (70, 172, 669)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 36 237582 828658 691212 438746 077575 043823 182285 124552 663829 971946 300976 469897 690254 645339 951806 661451 007120 > 4¹⁷² [i]