MinT - Best Known (163−105, 163, s)-Nets in Base 4

Best Known (163−105, 163, s)-Nets in Base 4

(163−105, 163, 66)-Net over F₄ — Constructive and digital

Digital (58, 163, 66)-net over F₄, using

t-expansion [i] based on digital (49, 163, 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]

(163−105, 163, 91)-Net over F₄ — Digital

Digital (58, 163, 91)-net over F₄, using

t-expansion [i] based on digital (50, 163, 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]

(163−105, 163, 464)-Net in Base 4 — Upper bound on s

There is no (58, 163, 465)-net in base 4, because

1 times m-reduction [i] would yield (58, 162, 465)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 34 840078 022808 075467 170018 186612 557636 393644 610437 156280 625927 154407 664003 789423 557727 017247 439760 > 4¹⁶² [i]