MinT - Best Known (51, 51+61, s)-Nets in Base 4

Best Known (51, 51+61, s)-Nets in Base 4

(51, 51+61, 66)-Net over F₄ — Constructive and digital

Digital (51, 112, 66)-net over F₄, using

t-expansion [i] based on digital (49, 112, 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]

(51, 51+61, 91)-Net over F₄ — Digital

Digital (51, 112, 91)-net over F₄, using

t-expansion [i] based on digital (50, 112, 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]

(51, 51+61, 653)-Net in Base 4 — Upper bound on s

There is no (51, 112, 654)-net in base 4, because

1 times m-reduction [i] would yield (51, 111, 654)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 6 761734 370368 463636 799701 326191 564892 973696 787737 954236 792322 560768 > 4¹¹¹ [i]