MinT - Best Known (213−77, 213, s)-Nets in Base 3

Best Known (213−77, 213, s)-Nets in Base 3

(213−77, 213, 156)-Net over F₃ — Constructive and digital

Digital (136, 213, 156)-net over F₃, using

15 times m-reduction [i] based on digital (136, 228, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 114, 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]

(213−77, 213, 264)-Net over F₃ — Digital

Digital (136, 213, 264)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(213−77, 213, 3410)-Net in Base 3 — Upper bound on s

There is no (136, 213, 3411)-net in base 3, because

1 times m-reduction [i] would yield (136, 212, 3411)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 141175 057802 808578 213242 971421 879862 804441 907514 549873 305806 617163 402518 698646 136813 728511 131464 321405 > 3²¹² [i]