MinT - Best Known (55, 168, s)-Nets in Base 4

Best Known (55, 168, s)-Nets in Base 4

(55, 168, 66)-Net over F₄ — Constructive and digital

Digital (55, 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]

(55, 168, 91)-Net over F₄ — Digital

Digital (55, 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]

(55, 168, 407)-Net in Base 4 — Upper bound on s

There is no (55, 168, 408)-net in base 4, because

1 times m-reduction [i] would yield (55, 167, 408)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 37951 986727 533656 348416 212503 546872 576760 848522 970360 441895 957441 808660 979028 044884 701396 762017 714800 > 4¹⁶⁷ [i]