MinT - Best Known (213−83, 213, s)-Nets in Base 3

Best Known (213−83, 213, s)-Nets in Base 3

(213−83, 213, 156)-Net over F₃ — Constructive and digital

Digital (130, 213, 156)-net over F₃, using

3 times m-reduction [i] based on digital (130, 216, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 108, 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]

(213−83, 213, 213)-Net over F₃ — Digital

Digital (130, 213, 213)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(213−83, 213, 2325)-Net in Base 3 — Upper bound on s

There is no (130, 213, 2326)-net in base 3, because

1 times m-reduction [i] would yield (130, 212, 2326)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 142564 224851 745877 263931 122425 627864 131068 818530 237423 654227 299668 096792 873718 306809 749828 991936 770909 > 3²¹² [i]