MinT - Best Known (248−81, 248, s)-Nets in Base 3

Best Known (248−81, 248, s)-Nets in Base 3

(248−81, 248, 168)-Net over F₃ — Constructive and digital

Digital (167, 248, 168)-net over F₃, using

3¹ times duplication [i] based on digital (166, 247, 168)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 51, 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 (115, 196, 148)-net over F₃, using
    - trace code for nets [i] based on digital (17, 98, 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]

(248−81, 248, 409)-Net over F₃ — Digital

Digital (167, 248, 409)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(248−81, 248, 6926)-Net in Base 3 — Upper bound on s

There is no (167, 248, 6927)-net in base 3, because

1 times m-reduction [i] would yield (167, 247, 6927)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7077 957061 109799 701276 960886 061243 818209 253905 153980 081548 000494 143283 959865 422398 210842 448180 742902 773541 186958 334129 > 3²⁴⁷ [i]