MinT - Best Known (82, 82+63, s)-Nets in Base 3

Best Known (82, 82+63, s)-Nets in Base 3

(82, 82+63, 80)-Net over F₃ — Constructive and digital

Digital (82, 145, 80)-net over F₃, using

3 times m-reduction [i] based on digital (82, 148, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 74, 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]

(82, 82+63, 113)-Net over F₃ — Digital

Digital (82, 145, 113)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(82, 82+63, 991)-Net in Base 3 — Upper bound on s

There is no (82, 145, 992)-net in base 3, because

1 times m-reduction [i] would yield (82, 144, 992)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 514 688881 379023 596827 247789 480019 306391 680885 813052 315609 391796 993665 > 3¹⁴⁴ [i]