MinT - Best Known (215−83, 215, s)-Nets in Base 3

Best Known (215−83, 215, s)-Nets in Base 3

(215−83, 215, 156)-Net over F₃ — Constructive and digital

Digital (132, 215, 156)-net over F₃, using

5 times m-reduction [i] based on digital (132, 220, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 110, 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]

(215−83, 215, 220)-Net over F₃ — Digital

Digital (132, 215, 220)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(215−83, 215, 2455)-Net in Base 3 — Upper bound on s

There is no (132, 215, 2456)-net in base 3, because

1 times m-reduction [i] would yield (132, 214, 2456)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 277779 623289 604922 613147 663196 659109 978454 628185 983699 262484 587702 653540 915248 199602 762754 103147 137457 > 3²¹⁴ [i]