MinT - Best Known (138, 138+55, s)-Nets in Base 3

Best Known (138, 138+55, s)-Nets in Base 3

(138, 138+55, 264)-Net over F₃ — Constructive and digital

Digital (138, 193, 264)-net over F₃, using

3¹ times duplication [i] based on digital (137, 192, 264)-net over F₃, using
- trace code for nets [i] based on digital (9, 64, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]

(138, 138+55, 509)-Net over F₃ — Digital

Digital (138, 193, 509)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+55, 13470)-Net in Base 3 — Upper bound on s

There is no (138, 193, 13471)-net in base 3, because

1 times m-reduction [i] would yield (138, 192, 13471)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 40 518027 283287 419892 284050 751144 082320 718401 536534 315884 964266 886981 099048 327855 554039 369291 > 3¹⁹² [i]