MinT - Best Known (65−33, 65, s)-Nets in Base 8

Best Known (65−33, 65, s)-Nets in Base 8

(65−33, 65, 82)-Net over F₈ — Constructive and digital

Digital (32, 65, 82)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 18, 17)-net over F₈, using
  - net from sequence [i] based on digital (2, 16)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
      - a function field by SÃ©mirat [i]
2. digital (14, 47, 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]

(65−33, 65, 118)-Net over F₈ — Digital

Digital (32, 65, 118)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(65−33, 65, 3969)-Net in Base 8 — Upper bound on s

There is no (32, 65, 3970)-net in base 8, because

1 times m-reduction [i] would yield (32, 64, 3970)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6290 581925 534046 538474 888370 770120 859054 078869 252748 367880 > 8⁶⁴ [i]