MinT - Best Known (177−121, 177, s)-Nets in Base 4

Best Known (177−121, 177, s)-Nets in Base 4

(177−121, 177, 66)-Net over F₄ — Constructive and digital

Digital (56, 177, 66)-net over F₄, using

t-expansion [i] based on digital (49, 177, 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]

(177−121, 177, 91)-Net over F₄ — Digital

Digital (56, 177, 91)-net over F₄, using

t-expansion [i] based on digital (50, 177, 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]

(177−121, 177, 403)-Net in Base 4 — Upper bound on s

There is no (56, 177, 404)-net in base 4, because

1 times m-reduction [i] would yield (56, 176, 404)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9744 895666 691745 049542 550293 406755 881204 845327 730024 535808 598331 626682 210561 717249 132545 729640 086448 457120 > 4¹⁷⁶ [i]