MinT - Best Known (172, 172+53, s)-Nets in Base 3

Best Known (172, 172+53, s)-Nets in Base 3

(172, 172+53, 464)-Net over F₃ — Constructive and digital

Digital (172, 225, 464)-net over F₃, using

3¹ times duplication [i] based on digital (171, 224, 464)-net over F₃, using
- t-expansion [i] based on digital (170, 224, 464)-net over F₃, using
  - trace code for nets [i] based on digital (2, 56, 116)-net over F₈₁, using
    - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
        a function field by SÃ©mirat [i]

(172, 172+53, 1192)-Net over F₃ — Digital

Digital (172, 225, 1192)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(172, 172+53, 68027)-Net in Base 3 — Upper bound on s

There is no (172, 225, 68028)-net in base 3, because

1 times m-reduction [i] would yield (172, 224, 68028)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 75036 455659 459190 864368 970797 785247 187626 126362 239992 217782 429430 708626 074019 846603 409272 904431 979398 490505 > 3²²⁴ [i]