MinT - Best Known (159, 159+87, s)-Nets in Base 3

Best Known (159, 159+87, s)-Nets in Base 3

(159, 159+87, 162)-Net over F₃ — Constructive and digital

Digital (159, 246, 162)-net over F₃, using

t-expansion [i] based on digital (157, 246, 162)-net over F₃, using
- 4 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]

(159, 159+87, 319)-Net over F₃ — Digital

Digital (159, 246, 319)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(159, 159+87, 4372)-Net in Base 3 — Upper bound on s

There is no (159, 246, 4373)-net in base 3, because

1 times m-reduction [i] would yield (159, 245, 4373)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 788 061044 080805 213404 438371 440848 546524 978716 756131 905097 597753 338029 696721 727173 603650 132119 552232 339483 279443 433291 > 3²⁴⁵ [i]