MinT - Best Known (248−79, 248, s)-Nets in Base 3

Best Known (248−79, 248, s)-Nets in Base 3

(248−79, 248, 176)-Net over F₃ — Constructive and digital

Digital (169, 248, 176)-net over F₃, using

1 times m-reduction [i] based on digital (169, 249, 176)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 55, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
        an explicitly constructive algebraic function field [i]
  2. digital (114, 194, 148)-net over F₃, using
    - trace code for nets [i] based on digital (17, 97, 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]

(248−79, 248, 442)-Net over F₃ — Digital

Digital (169, 248, 442)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(248−79, 248, 8055)-Net in Base 3 — Upper bound on s

There is no (169, 248, 8056)-net in base 3, because

1 times m-reduction [i] would yield (169, 247, 8056)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7076 788562 203746 632210 845786 522998 018551 336722 851011 026449 710723 690566 215892 473959 853075 492078 408035 930672 089180 113057 > 3²⁴⁷ [i]