MinT - Best Known (71, 164, s)-Nets in Base 8

Best Known (71, 164, s)-Nets in Base 8

(71, 164, 113)-Net over F₈ — Constructive and digital

Digital (71, 164, 113)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 57, 48)-net over F₈, using
  - net from sequence [i] based on digital (11, 47)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 11 and N(F) ≥ 48, using
      - a function field by SÃ©mirat [i]
2. digital (14, 107, 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]

(71, 164, 159)-Net over F₈ — Digital

Digital (71, 164, 159)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(71, 164, 162)-Net in Base 8

(71, 164, 162)-net in base 8, using

base change [i] based on digital (30, 123, 162)-net over F₁₆, using
- net from sequence [i] based on digital (30, 161)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 30 and N(F) ≥ 162, using
    - a curve from the manYPoints database [i]

(71, 164, 4046)-Net in Base 8 — Upper bound on s

There is no (71, 164, 4047)-net in base 8, because

1 times m-reduction [i] would yield (71, 163, 4047)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1602 337799 376062 363288 349659 808914 477539 201089 955664 996013 651246 468290 706250 294969 785440 758647 689551 262795 626100 807477 375367 807958 026080 532201 701650 > 8¹⁶³ [i]