MinT - Best Known (167−61, 167, s)-Nets in Base 3

Best Known (167−61, 167, s)-Nets in Base 3

(167−61, 167, 156)-Net over F₃ — Constructive and digital

Digital (106, 167, 156)-net over F₃, using

1 times m-reduction [i] based on digital (106, 168, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 84, 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]

(167−61, 167, 207)-Net over F₃ — Digital

Digital (106, 167, 207)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(167−61, 167, 2599)-Net in Base 3 — Upper bound on s

There is no (106, 167, 2600)-net in base 3, because

1 times m-reduction [i] would yield (106, 166, 2600)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 15 949753 058730 305799 898984 423738 087510 055022 196905 221668 088762 656266 722116 497009 > 3¹⁶⁶ [i]