MinT - Best Known (136−71, 136, s)-Nets in Base 8

Best Known (136−71, 136, s)-Nets in Base 8

(136−71, 136, 130)-Net over F₈ — Constructive and digital

Digital (65, 136, 130)-net over F₈, using

3 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)

(136−71, 136, 188)-Net over F₈ — Digital

Digital (65, 136, 188)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(136−71, 136, 6024)-Net in Base 8 — Upper bound on s

There is no (65, 136, 6025)-net in base 8, because

1 times m-reduction [i] would yield (65, 135, 6025)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 82 801111 221896 239757 892121 410419 959982 104279 112007 748658 065613 139753 316522 463462 095760 906326 568316 006167 253475 050616 122528 > 8¹³⁵ [i]