MinT - Best Known (68, 167, s)-Nets in Base 4

Best Known (68, 167, s)-Nets in Base 4

(68, 167, 66)-Net over F₄ — Constructive and digital

Digital (68, 167, 66)-net over F₄, using

t-expansion [i] based on digital (49, 167, 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]

(68, 167, 99)-Net over F₄ — Digital

Digital (68, 167, 99)-net over F₄, using

t-expansion [i] based on digital (61, 167, 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]

(68, 167, 658)-Net in Base 4 — Upper bound on s

There is no (68, 167, 659)-net in base 4, because

1 times m-reduction [i] would yield (68, 166, 659)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9078 775545 203934 005525 854898 183791 934788 183332 964558 001017 250201 493475 365771 561548 127661 778057 421832 > 4¹⁶⁶ [i]