MinT - Best Known (239−81, 239, s)-Nets in Base 3

Best Known (239−81, 239, s)-Nets in Base 3

(239−81, 239, 162)-Net over F₃ — Constructive and digital

Digital (158, 239, 162)-net over F₃, using

t-expansion [i] based on digital (157, 239, 162)-net over F₃, using
- 11 times m-reduction [i] based on digital (157, 250, 162)-net over F₃, using
  - trace code for nets [i] based on digital (32, 125, 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]

(239−81, 239, 353)-Net over F₃ — Digital

Digital (158, 239, 353)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(239−81, 239, 5401)-Net in Base 3 — Upper bound on s

There is no (158, 239, 5402)-net in base 3, because

1 times m-reduction [i] would yield (158, 238, 5402)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 361342 772385 732333 512296 569235 847743 714232 515684 077727 842636 755399 993460 899165 198859 469556 116231 239526 793439 386385 > 3²³⁸ [i]