MinT - Best Known (134−71, 134, s)-Nets in Base 4

Best Known (134−71, 134, s)-Nets in Base 4

(134−71, 134, 66)-Net over F₄ — Constructive and digital

Digital (63, 134, 66)-net over F₄, using

t-expansion [i] based on digital (49, 134, 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]

(134−71, 134, 99)-Net over F₄ — Digital

Digital (63, 134, 99)-net over F₄, using

t-expansion [i] based on digital (61, 134, 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]

(134−71, 134, 871)-Net in Base 4 — Upper bound on s

There is no (63, 134, 872)-net in base 4, because

1 times m-reduction [i] would yield (63, 133, 872)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 122 726466 937457 851382 817136 619979 754452 067289 464701 293701 906013 354197 799003 080291 > 4¹³³ [i]