MinT - Best Known (170−115, 170, s)-Nets in Base 4

Best Known (170−115, 170, s)-Nets in Base 4

(170−115, 170, 66)-Net over F₄ — Constructive and digital

Digital (55, 170, 66)-net over F₄, using

t-expansion [i] based on digital (49, 170, 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]

(170−115, 170, 91)-Net over F₄ — Digital

Digital (55, 170, 91)-net over F₄, using

t-expansion [i] based on digital (50, 170, 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]

(170−115, 170, 403)-Net in Base 4 — Upper bound on s

There is no (55, 170, 404)-net in base 4, because

1 times m-reduction [i] would yield (55, 169, 404)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 600057 607115 975780 576217 053836 167930 988994 470667 991835 458644 342159 655924 800008 867334 333813 424731 477200 > 4¹⁶⁹ [i]