MinT - Best Known (27, 27+15, s)-Nets in Base 3

Best Known (27, 27+15, s)-Nets in Base 3

(27, 27+15, 68)-Net over F₃ — Constructive and digital

Digital (27, 42, 68)-net over F₃, using

trace code for nets [i] based on digital (6, 21, 34)-net over F₉, using
- net from sequence [i] based on digital (6, 33)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 6 and N(F) ≥ 34, using
    - a function field by SÃ©mirat [i]

(27, 27+15, 82)-Net over F₃ — Digital

Digital (27, 42, 82)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(27, 27+15, 1046)-Net in Base 3 — Upper bound on s

There is no (27, 42, 1047)-net in base 3, because

1 times m-reduction [i] would yield (27, 41, 1047)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 36 574022 515730 436051 > 3⁴¹ [i]