MinT - Best Known (69, 69+29, s)-Nets in Base 8

Best Known (69, 69+29, s)-Nets in Base 8

(69, 69+29, 416)-Net over F₈ — Constructive and digital

Digital (69, 98, 416)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (20, 34, 208)-net over F₈, using
  - trace code for nets [i] based on digital (3, 17, 104)-net over F₆₄, using
    - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]
2. digital (35, 64, 208)-net over F₈, using
  - trace code for nets [i] based on digital (3, 32, 104)-net over F₆₄, using
    - net from sequence [i] based on digital (3, 103)-sequence over F₆₄ (see above)

(69, 69+29, 576)-Net in Base 8 — Constructive

(69, 98, 576)-net in base 8, using

6 times m-reduction [i] based on (69, 104, 576)-net in base 8, using
- trace code for nets [i] based on (17, 52, 288)-net in base 64, using
  - 4 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
    - base change [i] based on digital (9, 48, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(69, 69+29, 2351)-Net over F₈ — Digital

Digital (69, 98, 2351)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 69+29, 1561280)-Net in Base 8 — Upper bound on s

There is no (69, 98, 1561281)-net in base 8, because

1 times m-reduction [i] would yield (69, 97, 1561281)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3978 619016 122182 136509 979981 955313 181266 878232 379563 692590 993878 250767 577721 312660 317536 > 8⁹⁷ [i]