MinT - Best Known (197−131, 197, s)-Nets in Base 4

Best Known (197−131, 197, s)-Nets in Base 4

(197−131, 197, 66)-Net over F₄ — Constructive and digital

Digital (66, 197, 66)-net over F₄, using

t-expansion [i] based on digital (49, 197, 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]

(197−131, 197, 99)-Net over F₄ — Digital

Digital (66, 197, 99)-net over F₄, using

t-expansion [i] based on digital (61, 197, 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]

(197−131, 197, 494)-Net in Base 4 — Upper bound on s

There is no (66, 197, 495)-net in base 4, because

1 times m-reduction [i] would yield (66, 196, 495)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11297 325477 235735 900729 766528 104067 595525 940314 984146 876641 607796 272036 700428 684503 298646 776575 722994 387986 064401 497062 > 4¹⁹⁶ [i]