MinT - Best Known (60−31, 60, s)-Nets in Base 8

Best Known (60−31, 60, s)-Nets in Base 8

(60−31, 60, 74)-Net over F₈ — Constructive and digital

Digital (29, 60, 74)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 15, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [i]
2. digital (14, 45, 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]

(60−31, 60, 76)-Net in Base 8 — Constructive

(29, 60, 76)-net in base 8, using

base change [i] based on digital (5, 36, 76)-net over F₃₂, using
- net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
    - a function field by SÃ©mirat [i]

(60−31, 60, 103)-Net over F₈ — Digital

Digital (29, 60, 103)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(60−31, 60, 3262)-Net in Base 8 — Upper bound on s

There is no (29, 60, 3263)-net in base 8, because

1 times m-reduction [i] would yield (29, 59, 3263)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 191619 074964 858755 917901 012257 971335 955075 769519 786136 > 8⁵⁹ [i]