MinT - Best Known (65, 65+133, s)-Nets in Base 4

Best Known (65, 65+133, s)-Nets in Base 4

(65, 65+133, 66)-Net over F₄ — Constructive and digital

Digital (65, 198, 66)-net over F₄, using

t-expansion [i] based on digital (49, 198, 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]

(65, 65+133, 99)-Net over F₄ — Digital

Digital (65, 198, 99)-net over F₄, using

t-expansion [i] based on digital (61, 198, 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]

(65, 65+133, 478)-Net in Base 4 — Upper bound on s

There is no (65, 198, 479)-net in base 4, because

1 times m-reduction [i] would yield (65, 197, 479)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 43246 453599 777300 506457 438527 259355 739685 963865 710053 910935 975677 840243 783527 908861 145924 812382 911490 454893 404841 855277 > 4¹⁹⁷ [i]