MinT - Best Known (135, 135+85, s)-Nets in Base 3

Best Known (135, 135+85, s)-Nets in Base 3

(135, 135+85, 156)-Net over F₃ — Constructive and digital

Digital (135, 220, 156)-net over F₃, using

6 times m-reduction [i] based on digital (135, 226, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 113, 78)-net over F₉, using
  - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(135, 135+85, 224)-Net over F₃ — Digital

Digital (135, 220, 224)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(135, 135+85, 2497)-Net in Base 3 — Upper bound on s

There is no (135, 220, 2498)-net in base 3, because

1 times m-reduction [i] would yield (135, 219, 2498)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 311 165238 800132 253923 405653 128803 440064 648657 442594 661753 146411 905323 189980 256885 268691 930826 292494 000677 > 3²¹⁹ [i]