MinT - Best Known (52, 139, s)-Nets in Base 4

Best Known (52, 139, s)-Nets in Base 4

(52, 139, 66)-Net over F₄ — Constructive and digital

Digital (52, 139, 66)-net over F₄, using

t-expansion [i] based on digital (49, 139, 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]

(52, 139, 91)-Net over F₄ — Digital

Digital (52, 139, 91)-net over F₄, using

t-expansion [i] based on digital (50, 139, 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]

(52, 139, 447)-Net in Base 4 — Upper bound on s

There is no (52, 139, 448)-net in base 4, because

1 times m-reduction [i] would yield (52, 138, 448)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 132052 756526 762561 204150 556313 325678 325153 411505 225191 483955 229332 523198 813082 300775 > 4¹³⁸ [i]