MinT - Best Known (215−57, 215, s)-Nets in Base 3

Best Known (215−57, 215, s)-Nets in Base 3

(215−57, 215, 288)-Net over F₃ — Constructive and digital

Digital (158, 215, 288)-net over F₃, using

t-expansion [i] based on digital (157, 215, 288)-net over F₃, using
- 4 times m-reduction [i] based on digital (157, 219, 288)-net over F₃, using
  - trace code for nets [i] based on digital (11, 73, 96)-net over F₂₇, using
    - net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
        an explicitly constructive algebraic function field [i]

(215−57, 215, 721)-Net over F₃ — Digital

Digital (158, 215, 721)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(215−57, 215, 25007)-Net in Base 3 — Upper bound on s

There is no (158, 215, 25008)-net in base 3, because

1 times m-reduction [i] would yield (158, 214, 25008)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 271359 784948 783157 216573 961600 130089 550236 244047 760894 948568 908076 022892 556443 411647 798831 969398 828673 > 3²¹⁴ [i]