MinT - Best Known (157−51, 157, s)-Nets in Base 3

Best Known (157−51, 157, s)-Nets in Base 3

(157−51, 157, 156)-Net over F₃ — Constructive and digital

Digital (106, 157, 156)-net over F₃, using

11 times m-reduction [i] based on digital (106, 168, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 84, 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]

(157−51, 157, 281)-Net over F₃ — Digital

Digital (106, 157, 281)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(157−51, 157, 4804)-Net in Base 3 — Upper bound on s

There is no (106, 157, 4805)-net in base 3, because

1 times m-reduction [i] would yield (106, 156, 4805)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 270 733477 269284 560198 536590 963516 818092 542247 386579 875377 225359 395866 492587 > 3¹⁵⁶ [i]