MinT - Best Known (65, 65+69, s)-Nets in Base 8

Best Known (65, 65+69, s)-Nets in Base 8

(65, 65+69, 130)-Net over F₈ — Constructive and digital

Digital (65, 134, 130)-net over F₈, using

5 times m-reduction [i] based on digital (65, 139, 130)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (14, 51, 65)-net over F₈, using
    - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
        the Suzuki function field over F₈ [i]
  2. digital (14, 88, 65)-net over F₈, using
    - net from sequence [i] based on digital (14, 64)-sequence over F₈ (see above)

(65, 65+69, 197)-Net over F₈ — Digital

Digital (65, 134, 197)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(65, 65+69, 6571)-Net in Base 8 — Upper bound on s

There is no (65, 134, 6572)-net in base 8, because

1 times m-reduction [i] would yield (65, 133, 6572)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 296420 209703 115988 856164 676074 961293 595418 373784 118796 781730 016886 136991 741124 414076 867973 752776 584100 703322 508219 462039 > 8¹³³ [i]