MinT - Best Known (159, 159+85, s)-Nets in Base 3

Best Known (159, 159+85, s)-Nets in Base 3

(159, 159+85, 162)-Net over F₃ — Constructive and digital

Digital (159, 244, 162)-net over F₃, using

t-expansion [i] based on digital (157, 244, 162)-net over F₃, using
- 6 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+85, 331)-Net over F₃ — Digital

Digital (159, 244, 331)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(159, 159+85, 4714)-Net in Base 3 — Upper bound on s

There is no (159, 244, 4715)-net in base 3, because

1 times m-reduction [i] would yield (159, 243, 4715)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 87 313152 263713 060553 691326 405514 540981 776463 824520 012425 575638 245418 813794 057010 734330 823414 696018 294256 145025 371109 > 3²⁴³ [i]