MinT - Best Known (217−83, 217, s)-Nets in Base 3

Best Known (217−83, 217, s)-Nets in Base 3

(217−83, 217, 156)-Net over F₃ — Constructive and digital

Digital (134, 217, 156)-net over F₃, using

7 times m-reduction [i] based on digital (134, 224, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 112, 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]

(217−83, 217, 228)-Net over F₃ — Digital

Digital (134, 217, 228)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(217−83, 217, 2593)-Net in Base 3 — Upper bound on s

There is no (134, 217, 2594)-net in base 3, because

1 times m-reduction [i] would yield (134, 216, 2594)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 11 606063 302152 161788 760282 303662 429275 371618 398605 424675 743759 489494 847168 901915 370933 802433 589139 196725 > 3²¹⁶ [i]