MinT - Best Known (69, 69+95, s)-Nets in Base 4

Best Known (69, 69+95, s)-Nets in Base 4

(69, 69+95, 66)-Net over F₄ — Constructive and digital

Digital (69, 164, 66)-net over F₄, using

t-expansion [i] based on digital (49, 164, 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+95, 99)-Net over F₄ — Digital

Digital (69, 164, 99)-net over F₄, using

t-expansion [i] based on digital (61, 164, 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+95, 711)-Net in Base 4 — Upper bound on s

There is no (69, 164, 712)-net in base 4, because

1 times m-reduction [i] would yield (69, 163, 712)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 137 351636 738048 636769 517405 532299 278669 725528 062690 577648 712453 524996 626938 787449 761576 925752 153560 > 4¹⁶³ [i]