MinT - Best Known (69, 69+127, s)-Nets in Base 4

Best Known (69, 69+127, s)-Nets in Base 4

(69, 69+127, 66)-Net over F₄ — Constructive and digital

Digital (69, 196, 66)-net over F₄, using

t-expansion [i] based on digital (49, 196, 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]

(69, 69+127, 99)-Net over F₄ — Digital

Digital (69, 196, 99)-net over F₄, using

t-expansion [i] based on digital (61, 196, 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]

(69, 69+127, 541)-Net in Base 4 — Upper bound on s

There is no (69, 196, 542)-net in base 4, because

1 times m-reduction [i] would yield (69, 195, 542)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2710 496036 189078 966752 362490 080441 649839 273423 859970 891449 634582 719802 445359 940049 989924 221360 423786 622076 379128 860160 > 4¹⁹⁵ [i]