MinT - Best Known (167−59, 167, s)-Nets in Base 3

Best Known (167−59, 167, s)-Nets in Base 3

(167−59, 167, 156)-Net over F₃ — Constructive and digital

Digital (108, 167, 156)-net over F₃, using

5 times m-reduction [i] based on digital (108, 172, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 86, 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]

(167−59, 167, 229)-Net over F₃ — Digital

Digital (108, 167, 229)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(167−59, 167, 3113)-Net in Base 3 — Upper bound on s

There is no (108, 167, 3114)-net in base 3, because

1 times m-reduction [i] would yield (108, 166, 3114)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 15 991292 157676 691145 854915 810394 344469 642974 059251 343158 699697 151251 967426 040893 > 3¹⁶⁶ [i]