MinT - Best Known (168−109, 168, s)-Nets in Base 4

Best Known (168−109, 168, s)-Nets in Base 4

(168−109, 168, 66)-Net over F₄ — Constructive and digital

Digital (59, 168, 66)-net over F₄, using

t-expansion [i] based on digital (49, 168, 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]

(168−109, 168, 91)-Net over F₄ — Digital

Digital (59, 168, 91)-net over F₄, using

t-expansion [i] based on digital (50, 168, 91)-net over F₄, using
- net from sequence [i] based on digital (50, 90)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 50 and N(F) ≥ 91, using
    - a curve from the manYPoints database [i]

(168−109, 168, 465)-Net in Base 4 — Upper bound on s

There is no (59, 168, 466)-net in base 4, because

1 times m-reduction [i] would yield (59, 167, 466)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 36945 651219 699508 050311 789332 123672 850523 230382 944169 101723 354386 332734 253639 183562 632088 489349 188000 > 4¹⁶⁷ [i]