MinT - Best Known (161, 161+77, s)-Nets in Base 3

Best Known (161, 161+77, s)-Nets in Base 3

(161, 161+77, 168)-Net over F₃ — Constructive and digital

Digital (161, 238, 168)-net over F₃, using

3¹ times duplication [i] based on digital (160, 237, 168)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 49, 20)-net over F₃, using
    - net from sequence [i] based on digital (11, 19)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 9, N(F)Â =Â 19, and 1 place with degree 3 [i] based on function field F/F₃ with g(F) = 9 and N(F) ≥ 19, using an explicitly constructive algebraic function field [i]
  2. digital (111, 188, 148)-net over F₃, using
    - trace code for nets [i] based on digital (17, 94, 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]

(161, 161+77, 406)-Net over F₃ — Digital

Digital (161, 238, 406)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(161, 161+77, 7066)-Net in Base 3 — Upper bound on s

There is no (161, 238, 7067)-net in base 3, because

1 times m-reduction [i] would yield (161, 237, 7067)-net in base 3, but
- 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]