MinT - Best Known (82, 82+37, s)-Nets in Base 3

Best Known (82, 82+37, s)-Nets in Base 3

(82, 82+37, 156)-Net over F₃ — Constructive and digital

Digital (82, 119, 156)-net over F₃, using

1 times m-reduction [i] based on digital (82, 120, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 60, 78)-net over F₉, using
  - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(82, 82+37, 256)-Net over F₃ — Digital

Digital (82, 119, 256)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(82, 82+37, 5051)-Net in Base 3 — Upper bound on s

There is no (82, 119, 5052)-net in base 3, because

1 times m-reduction [i] would yield (82, 118, 5052)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 200 231439 501852 778761 652549 396295 339412 985493 358504 106377 > 3¹¹⁸ [i]