MinT - Best Known (199−74, 199, s)-Nets in Base 3

Best Known (199−74, 199, s)-Nets in Base 3

(199−74, 199, 156)-Net over F₃ — Constructive and digital

Digital (125, 199, 156)-net over F₃, using

7 times m-reduction [i] based on digital (125, 206, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 103, 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]

(199−74, 199, 229)-Net over F₃ — Digital

Digital (125, 199, 229)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(199−74, 199, 2661)-Net in Base 3 — Upper bound on s

There is no (125, 199, 2662)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 89060 235840 089534 272334 858243 401990 126512 572042 520033 838455 399933 551108 762166 838791 129416 691045 > 3¹⁹⁹ [i]