MinT - Best Known (59, 172, s)-Nets in Base 4

Best Known (59, 172, s)-Nets in Base 4

(59, 172, 66)-Net over F₄ — Constructive and digital

Digital (59, 172, 66)-net over F₄, using

t-expansion [i] based on digital (49, 172, 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]

(59, 172, 91)-Net over F₄ — Digital

Digital (59, 172, 91)-net over F₄, using

t-expansion [i] based on digital (50, 172, 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]

(59, 172, 454)-Net in Base 4 — Upper bound on s

There is no (59, 172, 455)-net in base 4, because

1 times m-reduction [i] would yield (59, 171, 455)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9 779294 635562 730659 972048 259103 620593 398060 000975 408988 174293 027729 034537 535210 246699 746636 548147 855480 > 4¹⁷¹ [i]