MinT - Best Known (250−83, 250, s)-Nets in Base 3

Best Known (250−83, 250, s)-Nets in Base 3

(250−83, 250, 167)-Net over F₃ — Constructive and digital

Digital (167, 250, 167)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (9, 50, 19)-net over F₃, using
  - net from sequence [i] based on digital (9, 18)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [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 (117, 200, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 100, 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]

(250−83, 250, 391)-Net over F₃ — Digital

Digital (167, 250, 391)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(250−83, 250, 6335)-Net in Base 3 — Upper bound on s

There is no (167, 250, 6336)-net in base 3, because

1 times m-reduction [i] would yield (167, 249, 6336)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 63882 061849 189262 555161 965050 922819 140305 401974 524534 226957 089434 136898 788315 260373 395838 799989 493758 247968 641727 819137 > 3²⁴⁹ [i]