MinT - Best Known (147−77, 147, s)-Nets in Base 4

Best Known (147−77, 147, s)-Nets in Base 4

(147−77, 147, 66)-Net over F₄ — Constructive and digital

Digital (70, 147, 66)-net over F₄, using

t-expansion [i] based on digital (49, 147, 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]

(147−77, 147, 105)-Net over F₄ — Digital

Digital (70, 147, 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]

(147−77, 147, 999)-Net in Base 4 — Upper bound on s

There is no (70, 147, 1000)-net in base 4, because

1 times m-reduction [i] would yield (70, 146, 1000)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8182 577091 344241 992356 879869 309746 472939 072490 745820 215404 625132 045393 846604 051021 838021 > 4¹⁴⁶ [i]