MinT - Best Known (138−59, 138, s)-Nets in Base 3

Best Known (138−59, 138, s)-Nets in Base 3

(138−59, 138, 80)-Net over F₃ — Constructive and digital

Digital (79, 138, 80)-net over F₃, using

4 times m-reduction [i] based on digital (79, 142, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 71, 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]

(138−59, 138, 113)-Net over F₃ — Digital

Digital (79, 138, 113)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(138−59, 138, 1019)-Net in Base 3 — Upper bound on s

There is no (79, 138, 1020)-net in base 3, because

1 times m-reduction [i] would yield (79, 137, 1020)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 237674 652759 187768 395403 455317 827862 052595 955202 077511 869078 450393 > 3¹³⁷ [i]