MinT - Best Known (66, 66+125, s)-Nets in Base 4

Best Known (66, 66+125, s)-Nets in Base 4

(66, 66+125, 66)-Net over F₄ — Constructive and digital

Digital (66, 191, 66)-net over F₄, using

t-expansion [i] based on digital (49, 191, 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]

(66, 66+125, 99)-Net over F₄ — Digital

Digital (66, 191, 99)-net over F₄, using

t-expansion [i] based on digital (61, 191, 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]

(66, 66+125, 508)-Net in Base 4 — Upper bound on s

There is no (66, 191, 509)-net in base 4, because

1 times m-reduction [i] would yield (66, 190, 509)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 494682 096030 080558 970025 742698 360699 766231 347816 389278 339447 548910 230809 382680 842413 747067 738358 410629 640713 028720 > 4¹⁹⁰ [i]