MinT - Best Known (57, 57+67, s)-Nets in Base 8

Best Known (57, 57+67, s)-Nets in Base 8

(57, 57+67, 111)-Net over F₈ — Constructive and digital

Digital (57, 124, 111)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 43, 46)-net over F₈, using
  - net from sequence [i] based on digital (10, 45)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 9, N(F)Â =Â 45, and 1 place with degree 2 [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using an explicitly constructive algebraic function field [i]
2. digital (14, 81, 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]

(57, 57+67, 152)-Net over F₈ — Digital

Digital (57, 124, 152)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 57+67, 4347)-Net in Base 8 — Upper bound on s

There is no (57, 124, 4348)-net in base 8, because

1 times m-reduction [i] would yield (57, 123, 4348)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1205 372462 651044 916789 404351 747649 714272 671872 738033 961998 336115 981632 605027 129899 318931 413152 166933 205785 255589 > 8¹²³ [i]