MinT - Best Known (56, 56+63, s)-Nets in Base 8

Best Known (56, 56+63, s)-Nets in Base 8

(56, 56+63, 113)-Net over F₈ — Constructive and digital

Digital (56, 119, 113)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 42, 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, 77, 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]

(56, 56+63, 159)-Net over F₈ — Digital

Digital (56, 119, 159)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 56+63, 4839)-Net in Base 8 — Upper bound on s

There is no (56, 119, 4840)-net in base 8, because

1 times m-reduction [i] would yield (56, 118, 4840)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 36905 643344 252134 391395 634306 113656 686158 996955 217839 555536 051373 559306 060263 673584 774577 684275 168939 436308 > 8¹¹⁸ [i]