MinT - Best Known (58, 58+59, s)-Nets in Base 9

Best Known (58, 58+59, s)-Nets in Base 9

(58, 58+59, 128)-Net over F₉ — Constructive and digital

Digital (58, 117, 128)-net over F₉, using

5 times m-reduction [i] based on digital (58, 122, 128)-net over F₉, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 45, 64)-net over F₉, using
    - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
        a function field by GarcÃa and Quoos [i]
  2. digital (13, 77, 64)-net over F₉, using
    - net from sequence [i] based on digital (13, 63)-sequence over F₉ (see above)

(58, 58+59, 217)-Net over F₉ — Digital

Digital (58, 117, 217)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(58, 58+59, 9554)-Net in Base 9 — Upper bound on s

There is no (58, 117, 9555)-net in base 9, because

1 times m-reduction [i] would yield (58, 116, 9555)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 493 126669 864481 855498 872990 198661 654548 805183 630494 488992 185041 495194 254430 917378 330602 829970 632610 972087 738425 > 9¹¹⁶ [i]