MinT - Best Known (174−57, 174, s)-Nets in Base 3

Best Known (174−57, 174, s)-Nets in Base 3

(174−57, 174, 156)-Net over F₃ — Constructive and digital

Digital (117, 174, 156)-net over F₃, using

16 times m-reduction [i] based on digital (117, 190, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 95, 78)-net over F₉, using
  - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(174−57, 174, 299)-Net over F₃ — Digital

Digital (117, 174, 299)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(174−57, 174, 4983)-Net in Base 3 — Upper bound on s

There is no (117, 174, 4984)-net in base 3, because

1 times m-reduction [i] would yield (117, 173, 4984)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 34965 765103 182920 070907 234907 879702 252626 652759 235827 121350 602857 599837 387356 912065 > 3¹⁷³ [i]