MinT - Best Known (53, 53+61, s)-Nets in Base 4

Best Known (53, 53+61, s)-Nets in Base 4

(53, 53+61, 66)-Net over F₄ — Constructive and digital

Digital (53, 114, 66)-net over F₄, using

t-expansion [i] based on digital (49, 114, 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]

(53, 53+61, 91)-Net over F₄ — Digital

Digital (53, 114, 91)-net over F₄, using

t-expansion [i] based on digital (50, 114, 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]

(53, 53+61, 719)-Net in Base 4 — Upper bound on s

There is no (53, 114, 720)-net in base 4, because

1 times m-reduction [i] would yield (53, 113, 720)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 109 794997 462497 841839 637540 144611 031831 765592 353367 124616 998655 477924 > 4¹¹³ [i]