MinT - Best Known (54, 54+103, s)-Nets in Base 4

Best Known (54, 54+103, s)-Nets in Base 4

(54, 54+103, 66)-Net over F₄ — Constructive and digital

Digital (54, 157, 66)-net over F₄, using

t-expansion [i] based on digital (49, 157, 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]

(54, 54+103, 91)-Net over F₄ — Digital

Digital (54, 157, 91)-net over F₄, using

t-expansion [i] based on digital (50, 157, 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]

(54, 54+103, 418)-Net in Base 4 — Upper bound on s

There is no (54, 157, 419)-net in base 4, because

1 times m-reduction [i] would yield (54, 156, 419)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8391 694606 099344 376646 761904 325885 619046 835202 601060 002236 061174 624031 522194 305179 456487 511840 > 4¹⁵⁶ [i]