MinT - Best Known (245−83, 245, s)-Nets in Base 3

Best Known (245−83, 245, s)-Nets in Base 3

(245−83, 245, 162)-Net over F₃ — Constructive and digital

Digital (162, 245, 162)-net over F₃, using

t-expansion [i] based on digital (157, 245, 162)-net over F₃, using
- 5 times m-reduction [i] based on digital (157, 250, 162)-net over F₃, using
  - trace code for nets [i] based on digital (32, 125, 81)-net over F₉, using
    - net from sequence [i] based on digital (32, 80)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
        F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(245−83, 245, 361)-Net over F₃ — Digital

Digital (162, 245, 361)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(245−83, 245, 5535)-Net in Base 3 — Upper bound on s

There is no (162, 245, 5536)-net in base 3, because

1 times m-reduction [i] would yield (162, 244, 5536)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 261 963182 354965 424685 164383 094946 939192 164217 877660 962964 144298 485405 681878 698766 620569 023874 316042 757512 919200 431425 > 3²⁴⁴ [i]