MinT - Best Known (58, 58+123, s)-Nets in Base 4

Best Known (58, 58+123, s)-Nets in Base 4

(58, 58+123, 66)-Net over F₄ — Constructive and digital

Digital (58, 181, 66)-net over F₄, using

t-expansion [i] based on digital (49, 181, 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]

(58, 58+123, 91)-Net over F₄ — Digital

Digital (58, 181, 91)-net over F₄, using

t-expansion [i] based on digital (50, 181, 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]

(58, 58+123, 421)-Net in Base 4 — Upper bound on s

There is no (58, 181, 422)-net in base 4, because

1 times m-reduction [i] would yield (58, 180, 422)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 621217 962859 873896 831239 553332 898451 492782 501965 102281 350847 861909 173262 041722 464536 054762 531614 916935 631440 > 4¹⁸⁰ [i]