MinT - Best Known (156−89, 156, s)-Nets in Base 4

Best Known (156−89, 156, s)-Nets in Base 4

(156−89, 156, 66)-Net over F₄ — Constructive and digital

Digital (67, 156, 66)-net over F₄, using

t-expansion [i] based on digital (49, 156, 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]

(156−89, 156, 99)-Net over F₄ — Digital

Digital (67, 156, 99)-net over F₄, using

t-expansion [i] based on digital (61, 156, 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]

(156−89, 156, 724)-Net in Base 4 — Upper bound on s

There is no (67, 156, 725)-net in base 4, because

1 times m-reduction [i] would yield (67, 155, 725)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2176 234600 102004 065989 723876 683560 980496 287148 871147 041182 469303 234214 128668 635933 616317 226388 > 4¹⁵⁵ [i]