MinT - Best Known (170−111, 170, s)-Nets in Base 4

Best Known (170−111, 170, s)-Nets in Base 4

(170−111, 170, 66)-Net over F₄ — Constructive and digital

Digital (59, 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]

(170−111, 170, 91)-Net over F₄ — Digital

Digital (59, 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]

(170−111, 170, 459)-Net in Base 4 — Upper bound on s

There is no (59, 170, 460)-net in base 4, because

1 times m-reduction [i] would yield (59, 169, 460)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 581120 854377 467958 194912 586671 629005 837972 562816 644288 909184 499618 820407 385392 634785 978829 546777 787960 > 4¹⁶⁹ [i]