MinT - Best Known (52, 52+31, s)-Nets in Base 3

Best Known (52, 52+31, s)-Nets in Base 3

(52, 52+31, 80)-Net over F₃ — Constructive and digital

Digital (52, 83, 80)-net over F₃, using

5 times m-reduction [i] based on digital (52, 88, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 44, 40)-net over F₉, using
  - net from sequence [i] based on digital (8, 39)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 8 and N(F) ≥ 40, using
      - a function field by SÃ©mirat [i]

(52, 52+31, 112)-Net over F₃ — Digital

Digital (52, 83, 112)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(52, 52+31, 1288)-Net in Base 3 — Upper bound on s

There is no (52, 83, 1289)-net in base 3, because

1 times m-reduction [i] would yield (52, 82, 1289)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1334 381786 488375 327053 990895 369048 312971 > 3⁸² [i]