MinT - Best Known (137, 137+53, s)-Nets in Base 3

Best Known (137, 137+53, s)-Nets in Base 3

(137, 137+53, 282)-Net over F₃ — Constructive and digital

Digital (137, 190, 282)-net over F₃, using

3¹ times duplication [i] based on digital (136, 189, 282)-net over F₃, using
- trace code for nets [i] based on digital (10, 63, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]

(137, 137+53, 541)-Net over F₃ — Digital

Digital (137, 190, 541)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+53, 15482)-Net in Base 3 — Upper bound on s

There is no (137, 190, 15483)-net in base 3, because

1 times m-reduction [i] would yield (137, 189, 15483)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 499556 253014 592457 465132 619628 330469 255438 064675 724028 032550 789805 560525 062736 941982 353317 > 3¹⁸⁹ [i]