MinT - Best Known (136, 136+69, s)-Nets in Base 3

Best Known (136, 136+69, s)-Nets in Base 3

(136, 136+69, 162)-Net over F₃ — Constructive and digital

Digital (136, 205, 162)-net over F₃, using

3 times m-reduction [i] based on digital (136, 208, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 104, 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]

(136, 136+69, 316)-Net over F₃ — Digital

Digital (136, 205, 316)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(136, 136+69, 4900)-Net in Base 3 — Upper bound on s

There is no (136, 205, 4901)-net in base 3, because

1 times m-reduction [i] would yield (136, 204, 4901)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 21 619179 698204 242472 134410 788200 618618 137134 550273 683051 536683 788396 202112 228588 773007 298560 741905 > 3²⁰⁴ [i]