MinT - Best Known (135, 135+59, s)-Nets in Base 3

Best Known (135, 135+59, s)-Nets in Base 3

(135, 135+59, 204)-Net over F₃ — Constructive and digital

Digital (135, 194, 204)-net over F₃, using

1 times m-reduction [i] based on digital (135, 195, 204)-net over F₃, using
- trace code for nets [i] based on digital (5, 65, 68)-net over F₂₇, using
  - net from sequence [i] based on digital (5, 67)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 5 and N(F) ≥ 68, using
      - a function field by SÃ©mirat [i]

(135, 135+59, 413)-Net over F₃ — Digital

Digital (135, 194, 413)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(135, 135+59, 8709)-Net in Base 3 — Upper bound on s

There is no (135, 194, 8710)-net in base 3, because

1 times m-reduction [i] would yield (135, 193, 8710)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 121 603195 773359 063179 619344 885629 729405 893899 799794 872175 407966 390722 023230 898144 529178 967989 > 3¹⁹³ [i]