MinT - Best Known (89, 89+41, s)-Nets in Base 3

Best Known (89, 89+41, s)-Nets in Base 3

(89, 89+41, 156)-Net over F₃ — Constructive and digital

Digital (89, 130, 156)-net over F₃, using

4 times m-reduction [i] based on digital (89, 134, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 67, 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]

(89, 89+41, 263)-Net over F₃ — Digital

Digital (89, 130, 263)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 89+41, 4943)-Net in Base 3 — Upper bound on s

There is no (89, 130, 4944)-net in base 3, because

1 times m-reduction [i] would yield (89, 129, 4944)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 35 496034 629928 404479 502910 203258 638526 090962 926231 508445 414273 > 3¹²⁹ [i]