MinT - Best Known (243−91, 243, s)-Nets in Base 3

Best Known (243−91, 243, s)-Nets in Base 3

(243−91, 243, 156)-Net over F₃ — Constructive and digital

Digital (152, 243, 156)-net over F₃, using

t-expansion [i] based on digital (147, 243, 156)-net over F₃, using
- 7 times m-reduction [i] based on digital (147, 250, 156)-net over F₃, using
  - trace code for nets [i] based on digital (22, 125, 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]

(243−91, 243, 268)-Net over F₃ — Digital

Digital (152, 243, 268)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(243−91, 243, 3199)-Net in Base 3 — Upper bound on s

There is no (152, 243, 3200)-net in base 3, because

1 times m-reduction [i] would yield (152, 242, 3200)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 29 421497 716104 226288 184136 533011 205902 748136 048699 793545 348064 829867 305564 874470 898433 338684 273987 911680 344945 029377 > 3²⁴² [i]