MinT - Best Known (135, 135+39, s)-Nets in Base 3

Best Known (135, 135+39, s)-Nets in Base 3

(135, 135+39, 600)-Net over F₃ — Constructive and digital

Digital (135, 174, 600)-net over F₃, using

3² times duplication [i] based on digital (133, 172, 600)-net over F₃, using
- trace code for nets [i] based on digital (4, 43, 150)-net over F₈₁, using
  - net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]

(135, 135+39, 1168)-Net over F₃ — Digital

Digital (135, 174, 1168)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(135, 135+39, 87580)-Net in Base 3 — Upper bound on s

There is no (135, 174, 87581)-net in base 3, because

1 times m-reduction [i] would yield (135, 173, 87581)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 34834 217669 065251 666365 658744 547994 301084 456998 690775 449463 035551 924233 290346 244811 > 3¹⁷³ [i]