MinT - Best Known (217−85, 217, s)-Nets in Base 3

Best Known (217−85, 217, s)-Nets in Base 3

(217−85, 217, 156)-Net over F₃ — Constructive and digital

Digital (132, 217, 156)-net over F₃, using

3 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]

(217−85, 217, 213)-Net over F₃ — Digital

Digital (132, 217, 213)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(217−85, 217, 2305)-Net in Base 3 — Upper bound on s

There is no (132, 217, 2306)-net in base 3, because

1 times m-reduction [i] would yield (132, 216, 2306)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 11 446157 136470 499804 872595 670511 513533 323700 576936 949894 191581 922795 540517 472595 180695 971671 982829 266853 > 3²¹⁶ [i]