MinT - Best Known (135, 135+65, s)-Nets in Base 3

Best Known (135, 135+65, s)-Nets in Base 3

(135, 135+65, 162)-Net over F₃ — Constructive and digital

Digital (135, 200, 162)-net over F₃, using

6 times m-reduction [i] based on digital (135, 206, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 103, 81)-net over F₉, using
  - net from sequence [i] based on digital (32, 80)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
      - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(135, 135+65, 345)-Net over F₃ — Digital

Digital (135, 200, 345)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(135, 135+65, 5897)-Net in Base 3 — Upper bound on s

There is no (135, 200, 5898)-net in base 3, because

1 times m-reduction [i] would yield (135, 199, 5898)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 88961 697873 747451 415366 923289 146557 350837 063858 275094 998531 667779 086994 154554 613050 196933 445441 > 3¹⁹⁹ [i]