MinT - Best Known (127, 127+77, s)-Nets in Base 3

Best Known (127, 127+77, s)-Nets in Base 3

(127, 127+77, 156)-Net over F₃ — Constructive and digital

Digital (127, 204, 156)-net over F₃, using

6 times m-reduction [i] based on digital (127, 210, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 105, 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]

(127, 127+77, 225)-Net over F₃ — Digital

Digital (127, 204, 225)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(127, 127+77, 2621)-Net in Base 3 — Upper bound on s

There is no (127, 204, 2622)-net in base 3, because

1 times m-reduction [i] would yield (127, 203, 2622)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7 272093 127989 879273 434044 290156 403308 665141 647963 406760 049539 667647 988748 862083 855467 197060 830181 > 3²⁰³ [i]