MinT - Best Known (237−76, 237, s)-Nets in Base 3

Best Known (237−76, 237, s)-Nets in Base 3

(237−76, 237, 172)-Net over F₃ — Constructive and digital

Digital (161, 237, 172)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (13, 51, 24)-net over F₃, using
  - net from sequence [i] based on digital (13, 23)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 13 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (110, 186, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 93, 74)-net over F₉, using
    - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
        a function field by GarcÃa and Quoos [i]

(237−76, 237, 416)-Net over F₃ — Digital

Digital (161, 237, 416)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(237−76, 237, 7066)-Net in Base 3 — Upper bound on s

There is no (161, 237, 7067)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 119867 541254 048429 593236 994974 986992 067712 552661 968127 525310 178850 893632 431731 963062 795954 120250 278666 735081 721549 > 3²³⁷ [i]